NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progression is presented here for the benefit of the students preparing for the board examination. It is very important for the students to get well versed with these solutions of NCERT to get a good score in class 10 examination. These NCERT problems are solved by experts at BYJUâ€™S. These solutions will help you understand and master different types of questions on arithmetic progressions. NCERT solutions help you to attain perfection in solving different kinds of questions.
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Class 10 Maths Chapter 5 Exercise 5.1 Page: 99
1. In which of the following situations, does the list of numbers involved make arithmetic progression and why?
(i) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each additional km.
Solution:
We can write the given condition as;
Taxi fare for 1 km = 15
Taxi fare for first 2 kms = 15 + 8 = 23
Taxi fare for first 3 kms = 23 + 8 = 31
Taxi fare for first 4 kms = 31 + 8 = 39
And so onâ€¦â€¦
Thus, 15, 23, 31, 39 â€¦ forms an A.P. because every next term is 8 more than the preceding term.
(ii) The amount of air present in a cylinder when a vacuum pump removes 1/4Â of the air remaining in the cylinder at a time.
Solution:
Let the volume of air in a cylinder, initially, beÂ VÂ litres.
In each stroke, the vacuum pump removesÂ 1/4thÂ of airÂ remaining in the cylinder at a time. Or we can say, after every stroke, 1 â€“ 1/4 = 3/4th part of air will remain.
Therefore, volumes will beÂ V, 3V/4 , (3V/4)^{2}Â , (3V/4)^{3}â€¦and so on
Clearly, we can see here, the adjacent terms of this series do not have the common difference between them. Therefore, this series is not an A.P.
(iii) The cost of digging a well after every metre of digging, when it costs Rs 150 for the first metre and rises by Rs 50 for each subsequent metre.
Solution:
We can write the given condition as;
Cost of digging a well for first metre = Rs.150
Cost of digging a well for first 2 metres = Rs.150 + 50 = Rs.200
Cost of digging a well for first 3 metres = Rs.200 + 50 = Rs.250
Cost of digging a well for first 4 metres =Rs.250 + 50 = Rs.300
And so on..
Clearly, 150, 200, 250, 300 â€¦ forms an A.P. with a common difference of 50 between each term.
(iv) The amount of money in the account every year, when Rs 10000 is deposited at compound interest at 8% per annum.
Solution:
2. Write first four terms of the A.P. when the first term a and the common differenced are given as follows:
(i)Â aÂ = 10,Â dÂ = 10
(ii)Â aÂ = -2,Â dÂ = 0
(iii)Â aÂ = 4,Â dÂ = â€“ 3
(iv)Â aÂ = -1Â dÂ = 1/2
(v)Â aÂ = â€“ 1.25,Â dÂ = â€“ 0.25
Solutions:
(i)Â aÂ = 10,Â dÂ = 10
Let us consider, the Arithmetic Progression series beÂ a_{1},Â a_{2},Â a_{3},Â a_{4},Â a_{5}Â â€¦
a_{1}Â =Â aÂ = 10
a_{2}Â =Â a_{1}Â +Â dÂ = 10 + 10 = 20
a_{3}Â =Â a_{2}Â +Â dÂ = 20 + 10 = 30
a_{4}Â =Â a_{3}Â +Â dÂ = 30 + 10 = 40
a_{5}Â =Â a_{4}Â +Â dÂ = 40 + 10 = 50
And so onâ€¦
Therefore, the A.P. series will be 10, 20, 30, 40, 50 â€¦
And First four terms of this A.P. will be 10, 20, 30, and 40.
(ii)Â aÂ = â€“ 2,Â dÂ = 0
Let us consider, the Arithmetic Progression series beÂ a_{1},Â a_{2},Â a_{3},Â a_{4},Â a_{5}Â â€¦
a_{1}Â =Â aÂ = -2
a_{2}Â =Â a_{1}Â +Â dÂ = â€“ 2 + 0 = â€“ 2
a_{3}Â =Â a_{2}Â + d = â€“ 2 + 0 = â€“ 2
a_{4}Â =Â a_{3}Â +Â dÂ = â€“ 2 + 0 = â€“ 2
Therefore, the A.P. series will be â€“ 2, â€“ 2, â€“ 2, â€“ 2 â€¦
And, First four terms of this A.P. will be â€“ 2, â€“ 2, â€“ 2 and â€“ 2.
(iii)Â aÂ = 4,Â dÂ = â€“ 3
Let us consider, the Arithmetic Progression series beÂ a_{1},Â a_{2},Â a_{3},Â a_{4},Â a_{5}Â â€¦
a_{1}Â =Â aÂ = 4
a_{2}Â =Â a_{1}Â +Â dÂ = 4 â€“ 3 = 1
a_{3}Â =Â a_{2}Â +Â dÂ = 1 â€“ 3 = â€“ 2
a_{4}Â =Â a_{3}Â +Â dÂ = â€“ 2 â€“ 3 = â€“ 5
Therefore, the A.P. series will be 4, 1, â€“ 2 â€“ 5 â€¦
And, First four terms of this A.P. will be 4, 1, â€“ 2 and â€“ 5.
(iv)Â aÂ = â€“ 1,Â dÂ = Â½
Let us consider, the Arithmetic Progression series beÂ a_{1},Â a_{2},Â a_{3},Â a_{4},Â a_{5}Â â€¦
a_{2}Â =Â a_{1}Â +Â dÂ = -1Â + 1/2 = -1/2
a_{3}Â =Â a_{2}Â +Â dÂ = -1/2Â + 1/2 = 0
a_{4}Â =Â a_{3}Â +Â dÂ = 0 + 1/2 = 1/2
Thus, the A.P. series will be-1, -1/2, 0, 1/2
And First four terms of this A.P. will be -1, -1/2, 0 and 1/2.
(v)Â aÂ = â€“ 1.25,Â dÂ = â€“ 0.25
Let us consider, the Arithmetic Progression series beÂ a_{1},Â a_{2},Â a_{3},Â a_{4},Â a_{5}Â â€¦
a_{1}Â =Â aÂ = â€“ 1.25
a_{2}Â =Â a_{1}Â +Â dÂ = â€“ 1.25 â€“ 0.25 = â€“ 1.50
a_{3}Â =Â a_{2}Â +Â dÂ = â€“ 1.50 â€“ 0.25 = â€“ 1.75
a_{4}Â =Â a_{3}Â +Â dÂ = â€“ 1.75 â€“ 0.25 = â€“ 2.00
Therefore, the series will be 1.25, â€“ 1.50, â€“ 1.75, â€“ 2.00 â€¦â€¦..
And first four terms of this A.P. will be â€“ 1.25, â€“ 1.50, â€“ 1.75 and â€“ 2.00.
3. For the following A.P.s, write the first term and the common difference.
(i) 3, 1, â€“ 1, â€“ 3 â€¦
(ii) -5, â€“ 1, 3, 7 â€¦
(iii) 1/3, 5/3, 9/3, 13/3 â€¦.
(iv) 0.6, 1.7, 2.8, 3.9 â€¦
Solutions
(i) Given series,
3, 1, â€“ 1, â€“ 3 â€¦
First term,Â aÂ = 3
And
Common difference,Â dÂ = Second term â€“ First term
â‡’ 1 â€“ 3 = â€“ 2
â‡’ d = -2
(ii) Given series, â€“ 5, â€“ 1, 3, 7 â€¦
First term,Â aÂ = â€“ 5
And
Common difference,Â dÂ = Second term â€“ First term
â‡’ ( â€“ 1) â€“ ( â€“ 5) = â€“ 1 + 5 = 4
(iii) Given series, 1/3, 5/3, 9/3, 13/3 â€¦.
First term,Â aÂ = 1/3
And
Common difference,Â dÂ = Second term â€“ First term
â‡’ 5/3 â€“ 1/3 = 4/3
(iv) Given series, 0.6, 1.7, 2.8, 3.9 â€¦
First term,Â aÂ = 0.6
And
Common difference,Â dÂ = Second term â€“ First term
â‡’ 1.7 â€“ 0.6
â‡’ 1.1
4. Which of the following are APs? If they form an A.P. find the common differenceÂ dÂ and write three more terms.
(i) 2, 4, 8, 16 â€¦
(ii) 2, 5/2, 3, 7/2 â€¦.
(iii) -1.2, -3.2, -5.2, -7.2 â€¦
(iv) -10, â€“ 6, â€“ 2, 2 â€¦
(v) 3, 3 +Â âˆš2, 3Â + 2âˆš2, 3Â + 3âˆš2
(vi) 0.2, 0.22, 0.222, 0.2222 â€¦.
(vii) 0, â€“ 4, â€“ 8, â€“ 12 â€¦
(viii) -1/2, -1/2, -1/2, -1/2 â€¦.
(ix) 1, 3, 9, 27 â€¦
(x)Â a, 2a, 3a, 4aÂ â€¦
(xi)Â a,Â a^{2},Â a^{3},Â a^{4}Â â€¦
(xii) âˆš2, âˆš8, âˆš18, âˆš32Â â€¦
(xiii) âˆš3, âˆš6, âˆš9, âˆš12Â â€¦
(xiv) 1^{2}, 3^{2}, 5^{2}, 7^{2}Â â€¦
(xv) 1^{2}, 5^{2}, 7^{2}, 7^{3}Â â€¦
Solutions
(i) Given to us,
2, 4, 8, 16 â€¦
Here, the common difference is;
a_{2}Â â€“Â a_{1}Â = 4 â€“ 2 = 2
a_{3}Â â€“Â a_{2}Â = 8 â€“ 4 = 4
a_{4}Â â€“Â a_{3}Â = 16 â€“ 8 = 8
Since,Â a_{n}_{+1}Â â€“Â a_{n}Â or the common difference is not the same every time.
Therefore, the given series are not forming an A.P.
(ii) Given, 2, 5/2, 3, 7/2 â€¦.
Here,
a_{2}Â â€“Â a_{1}Â =Â 5/2 â€“ 2 = Â½
a_{3}Â â€“Â a_{2}Â =Â 3 â€“ 5/2 = Â½
a_{4}Â â€“Â a_{3}Â =Â 7/2 â€“ 3 = Â½
Since, a_{n}_{+1}Â â€“Â a_{n}Â or the common difference is same every time.
Therefore,Â dÂ = 1/2Â and the given series are in A.P.
The next three terms are;
a_{5}Â = 7/2Â + 1/2 = 4
a_{6}Â = 4Â + 1/2 = 9/2
a_{7}Â = 9/2Â + 1/2 = 5
(iii)Â Given, -1.2, â€“ 3.2, -5.2, -7.2 â€¦
Here,
a_{2}Â â€“Â a_{1}Â = ( -3.2) â€“ ( -1.2) = -2
a_{3}Â â€“Â a_{2}Â = ( -5.2) â€“ ( -3.2) = -2
a_{4}Â â€“Â a_{3}Â = ( -7.2) â€“ ( -5.2) = -2
Since, a_{n}_{+1}Â â€“Â a_{n}Â or common difference is same every time.
Therefore,Â dÂ = -2 and the given series are in A.P.
Hence, next three terms are;
a_{5}Â = â€“ 7.2 â€“ 2 = â€“ 9.2
a_{6}Â = â€“ 9.2 â€“ 2 = â€“ 11.2
a_{7}Â = â€“ 11.2 â€“ 2 = â€“ 13.2
(iv) Given, -10, â€“ 6, â€“ 2, 2 â€¦
Here, the terms and their difference are;
a_{2}Â â€“Â a_{1}Â = (-6) â€“ (-10) = 4
a_{3}Â â€“Â a_{2}Â = (-2) â€“ (-6) = 4
a_{4}Â â€“Â a_{3}Â = (2) â€“ (-2) = 4
Since,Â a_{n}_{+1}Â â€“Â a_{n}Â or the common difference is same every time.
Therefore,Â dÂ = 4 and the given numbers are in A.P.
Hence, next three terms are;
a_{5}Â = 2 + 4 = 6
a_{6}Â = 6 + 4 = 10
a_{7}Â = 10 + 4 = 14
(v) Given, 3, 3 +Â âˆš2, 3Â + 2âˆš2, 3Â + 3âˆš2
Here,
a_{2}Â â€“Â a_{1}Â = 3 +Â âˆš2Â â€“ 3 = âˆš2
a_{3}Â â€“Â a_{2}Â = (3Â + 2âˆš2)Â â€“ (3 +Â âˆš2) = âˆš2
a_{4}Â â€“Â a_{3}Â = (3Â + 3âˆš2) â€“ (3Â + 2âˆš2) = âˆš2
Since,Â a_{n}_{+1}Â â€“Â a_{n}Â or the common difference is same every time.
Therefore,Â dÂ =Â âˆš2Â and the given series forms a A.P.
Hence, next three terms are;
a_{5}Â = (3 +Â âˆš2)Â + âˆš2Â = 3 + 4âˆš2
a_{6}Â = (3 + 4âˆš2) + âˆš2Â = 3 + 5âˆš2
a_{7}Â = (3 + 5âˆš2)Â + âˆš2Â = 3 + 6âˆš2
(vi)Â 0.2, 0.22, 0.222, 0.2222 â€¦.
Here,
a_{2}Â â€“Â a_{1}Â =Â 0.22 â€“ 0.2 = 0.02
a_{3}Â â€“Â a_{2}Â =Â 0.222 â€“ 0.22 = 0.002
a_{4}Â â€“Â a_{3}Â =Â 0.2222 â€“ 0.222 = 0.0002
Since,Â a_{n}_{+1}Â â€“Â a_{n}Â or the common difference is not same every time.
Therefore, and the given series doesnâ€™t forms a A.P.
(vii)Â 0, -4, -8, -12 â€¦
Here,
a_{2}Â â€“Â a_{1}Â =Â (-4) â€“ 0 = -4
a_{3}Â â€“Â a_{2}Â =Â (-8) â€“ (-4) = -4
a_{4}Â â€“Â a_{3}Â =Â (-12) â€“ (-8) = -4
Since,Â a_{n}_{+1}Â â€“Â a_{n}Â or the common difference is same every time.
Therefore,Â dÂ =Â -4Â and and the given series forms a A.P.
Hence, next three terms are;
a_{5}Â =Â -12 â€“ 4 = -16
a_{6}Â =Â -16 â€“ 4 = -20
a_{7}Â =Â -20 â€“ 4 = -24
(viii) -1/2, -1/2, -1/2, -1/2 â€¦.
Here,
a_{2}Â â€“Â a_{1}Â = (-1/2) â€“ (-1/2) = 0
a_{3}Â â€“Â a_{2}Â = (-1/2) â€“ (-1/2) = 0
a_{4}Â â€“Â a_{3}Â = (-1/2) â€“ (-1/2) = 0
Since,Â a_{n}_{+1}Â â€“Â a_{n}Â or the common difference is same every time.
Therefore,Â dÂ = 0 and and the given series forms a A.P.
Hence, next three terms are;
a_{5}Â = (-1/2) â€“ 0 = -1/2
a_{6}Â =Â (-1/2) â€“ 0 = -1/2
a_{7}Â =Â (-1/2) â€“ 0 = -1/2
(ix) 1, 3, 9, 27 â€¦
Here,
a_{2}Â â€“Â a_{1}Â =Â 3 â€“ 1 = 2
a_{3}Â â€“Â a_{2}Â =Â 9 â€“ 3 = 6
a_{4}Â â€“Â a_{3}Â =Â 27 â€“ 9 = 18
Since,Â a_{n}_{+1}Â â€“Â a_{n}Â or the common difference is not same every time.
Therefore, and the given series doesnâ€™t forms a A.P.
(x)Â a, 2a, 3a, 4aÂ â€¦
Here,
a_{2}Â â€“Â a_{1}Â =Â 2aÂ â€“Â aÂ =Â a
a_{3}Â â€“Â a_{2}Â =Â 3aÂ â€“ 2aÂ =Â a
a_{4}Â â€“Â a_{3}Â =Â 4aÂ â€“ 3aÂ =Â a
Since,Â a_{n}_{+1}Â â€“Â a_{n}Â or the common difference is same every time.
Therefore,Â dÂ =Â aÂ and and the given series forms a A.P.
Hence, next three terms are;
a_{5}Â =Â 4aÂ +Â aÂ = 5a
a_{6}Â = 5aÂ +Â aÂ = 6a
a_{7}Â =Â 6aÂ +Â aÂ = 7a
(xi)Â a,Â a^{2},Â a^{3},Â a^{4}Â â€¦
Here,
a_{2}Â â€“Â a_{1}Â =Â a^{2Â }â€“Â aÂ = (aÂ â€“ 1)
a_{3}Â â€“Â a_{2}Â =Â a^{3Â }â€“^{Â }a^{2Â }=Â a^{2Â }(aÂ â€“ 1)
a_{4}Â â€“Â a_{3}Â =Â a^{4}Â â€“Â a^{3Â }=Â a^{3}(aÂ â€“ 1)
Since,Â a_{n}_{+1}Â â€“Â a_{n}Â or the common difference is not same every time.
Therefore, the given series doesnâ€™t forms a A.P.
(xii) âˆš2, âˆš8, âˆš18, âˆš32Â â€¦
Here,
a_{2}Â â€“Â a_{1}Â = âˆš8Â â€“ âˆš2Â Â = 2âˆš2Â â€“ âˆš2Â = âˆš2
a_{3}Â â€“Â a_{2}Â = âˆš18Â â€“ âˆš8Â = 3âˆš2Â â€“ 2âˆš2Â = âˆš2
a_{4}Â â€“Â a_{3}Â = 4âˆš2Â â€“ 3âˆš2Â = âˆš2
Since,Â a_{n}_{+1}Â â€“Â a_{n}Â or the common difference is same every time.
Therefore,Â dÂ =Â âˆš2Â and the given series forms a A.P.
Hence, next three terms are;
a_{5}Â = âˆš32Â Â + âˆš2Â = 4âˆš2Â + âˆš2Â = 5âˆš2Â = âˆš50
a_{6}Â = 5âˆš2Â +âˆš2Â = 6âˆš2Â = âˆš72
a_{7}Â = 6âˆš2Â + âˆš2Â = 7âˆš2Â = âˆš98
(xiii)Â âˆš3, âˆš6, âˆš9, âˆš12Â â€¦
Here,
a_{2}Â â€“Â a_{1}Â = âˆš6Â â€“ âˆš3Â = âˆš3Â Ã— 2Â -âˆš3Â = âˆš3(âˆš2Â â€“ 1)
a_{3}Â â€“Â a_{2}Â = âˆš9Â â€“ âˆš6Â = 3 â€“ âˆš6Â = âˆš3(âˆš3Â â€“ âˆš2)
a_{4}Â â€“Â a_{3}Â = âˆš12Â â€“ âˆš9Â = 2âˆš3Â â€“ âˆš3Â Ã— 3 = âˆš3(2Â â€“ âˆš3)
Since,Â a_{n}_{+1}Â â€“Â a_{n}Â or the common difference is not same every time.
Therefore, the given series doesnâ€™t forms a A.P.
(xiv) 1^{2}, 3^{2}, 5^{2}, 7^{2}Â â€¦
Or, 1, 9, 25, 49 â€¦..
Here,
a_{2}Â âˆ’Â a_{1}Â = 9 âˆ’ 1 = 8
a_{3}Â âˆ’Â a_{2Â }= 25 âˆ’ 9 = 16
a_{4}Â âˆ’Â a_{3}Â = 49 âˆ’ 25 = 24
Since,Â a_{n}_{+1}Â â€“Â a_{n}Â or the common difference is not same every time.
Therefore, the given series doesnâ€™t forms a A.P.
(xv) 1^{2}, 5^{2}, 7^{2}, 73 â€¦
Or 1, 25, 49, 73 â€¦
Here,
a_{2}Â âˆ’Â a_{1}Â = 25 âˆ’ 1 = 24
a_{3}Â âˆ’Â a_{2Â }= 49 âˆ’ 25 = 24
a_{4}Â âˆ’Â a_{3}Â = 73 âˆ’ 49 = 24
Since,Â a_{n}_{+1}Â â€“Â a_{n}Â or the common difference is same every time.
Therefore,Â dÂ = 24Â and the given series forms a A.P.
Hence, next three terms are;
a_{5}Â = 73+ 24 = 97
a_{6}Â = 97 + 24 = 121
a_{7Â }= 121 + 24 = 145
Class 10 Maths Chapter 5 Exercise 5.2 Page: 105
1. Fill in the blanks in the following table, given thatÂ aÂ is the first term,Â dÂ the common difference andÂ a_{n}Â theÂ n^{th}Â term of the A.P.
a | d | n | a_{n} | |
(i) | 7 | 3 | 8 | â€¦â€¦ |
(ii) | âˆ’ 18 | â€¦.. | 10 | 0 |
(iii) | â€¦.. | âˆ’ 3 | 18 | âˆ’ 5 |
(iv) | âˆ’ 18.9 | 2.5 | â€¦.. | 3.6 |
(v) | 3.5 | 0 | 105 | â€¦.. |
Solutions:
(i)Given,
First term, aÂ = 7
Common difference,Â dÂ = 3
Number of terms,Â nÂ = 8,
We have to find the nth term, a_{n}Â = ?
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Putting the values,
â‡’ 7 + (8 âˆ’ 1) 3
â‡’ 7 + (7) 3
â‡’ 7 + 21 = 28
Hence,Â a_{n}Â = 28
(ii) Given,
First term, aÂ = -18
Common difference,Â dÂ = ?
Number of terms,Â nÂ = 10
Nth term, a_{n}Â = 0
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Putting the values,
0 = âˆ’ 18 + (10 âˆ’ 1)Â d
18 = 9d
dÂ = 18/9 = 2
Hence, common difference,Â dÂ = 2
(iii) Given,
First term, aÂ = ?
Common difference,Â dÂ = -3
Number of terms,Â nÂ = 18
Nth term, a_{n}Â = -5
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Putting the values,
âˆ’5 =Â aÂ + (18 âˆ’ 1) (âˆ’3)
âˆ’5 =Â aÂ + (17) (âˆ’3)
âˆ’5 =Â aÂ âˆ’ 51
aÂ = 51 âˆ’ 5 = 46
Hence,Â aÂ = 46
(iv)Â Given,
First term, aÂ = -18.9
Common difference,Â dÂ = 2.5
Number of terms,Â nÂ = ?
Nth term, a_{n}Â = 3.6
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Putting the values,
3.6 = âˆ’ 18.9 + (nÂ âˆ’ 1) 2.5
3.6 + 18.9 = (nÂ âˆ’ 1) 2.5
22.5 = (nÂ âˆ’ 1) 2.5
(nÂ â€“ 1) = 22.5/2.5
nÂ â€“ 1 = 9
nÂ = 10
Hence,Â nÂ = 10
(v)Â Given,
First term, aÂ = 3.5
Common difference,Â dÂ = 0
Number of terms,Â nÂ = 105
Nth term, a_{n}Â = ?
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Putting the values,
a_{n}Â = 3.5 + (105 âˆ’ 1) 0
a_{n}Â = 3.5 + 104 Ã— 0
a_{n}Â = 3.5
Hence,Â a_{n}Â = 3.5
2.Choose the correct choice in the following and justify:
(i) 30^{th}Â term of the A.P: 10, 7, 4, â€¦, is
(A)Â 97 (B)Â 77 (C)Â âˆ’77 (D) âˆ’87
(ii) 11^{thÂ }term of the A.P. -3, -1/2, ,2 â€¦. is
(A) 28 (B) 22 (C) â€“ 38 (D)
Solutions:
(i) Given here,
A.P. = 10, 7, 4, â€¦
Therefore, we can find,
First term,Â aÂ = 10
Common difference,Â dÂ =Â a_{2}Â âˆ’Â a_{1Â }= 7 âˆ’ 10 = âˆ’3
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Putting the values;
a_{30}Â = 10 + (30 âˆ’ 1) (âˆ’3)
a_{30}Â = 10 + (29) (âˆ’3)
a_{30}Â = 10 âˆ’ 87 = âˆ’77
Hence, the correct answer is optionÂ C.
(ii) Given here,
A.P. =Â -3, -1/2, ,2 â€¦
Therefore, we can find,
First termÂ aÂ = â€“ 3
Common difference,Â dÂ =Â a_{2}Â âˆ’Â a_{1}Â = (-1/2) â€“ (-3)
â‡’ (-1/2)Â + 3 = 5/2
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Putting the values;
a_{11}Â = 3Â + (11 -1)(5/2)
a_{11}Â = 3Â + (10)(5/2)
a_{11}Â = -3Â + 25
a_{11}Â = 22
Hence, the answer is option B.
3. In the following APs find the missing term in the boxes.
Solutions:
(i)Â For the given A.P., 2,2 , 26
The first and third term are;
aÂ = 2
a_{3}Â = 26
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Therefore, putting the values here,
a_{3}Â = 2 + (3 â€“ 1)Â d
26 = 2 + 2d
24 = 2d
dÂ = 12
a_{2}Â = 2 + (2 â€“ 1) 12
= 14
Therefore, 14 is the missing term.
(ii)Â For the given A.P., , 13, ,3
a_{2}Â = 13 and
a_{4}Â = 3
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Therefore, putting the values here,
a_{2}Â =Â aÂ + (2 â€“ 1)Â d
13 =Â aÂ +Â dÂ â€¦â€¦â€¦â€¦â€¦â€¦.Â (i)
a_{4}Â =Â aÂ + (4 â€“ 1)Â d
3 =Â aÂ + 3dÂ â€¦â€¦â€¦â€¦..Â (ii)
On subtracting equationÂ (i)Â fromÂ (ii), we get,
â€“ 10 = 2d
dÂ = â€“ 5
From equationÂ (i), putting the value of d,we get
13 =Â aÂ + (-5)
aÂ = 18
a_{3}Â = 18 + (3 â€“ 1) (-5)
= 18 + 2 (-5) = 18 â€“ 10 = 8
Therefore, the missing terms are 18 and 8 respectively.
(iii) For the given A.P.,
aÂ = 5 and
a_{4}Â = 19/2
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Therefore, putting the values here,
a_{4}Â =Â aÂ + (4 â€“ 1)Â d
19/2 =Â 5Â + 3d
19/2 â€“ 5 = 3d3d =Â 9/2
d = 3/2
a_{2}Â =Â aÂ + (2 â€“ 1)Â d
a_{2}Â =Â 5Â + 3/2
a_{2}Â =Â 13/2
a_{3}Â =Â aÂ + (3 â€“ 1)Â d
a_{3}Â =Â 5Â + 2Ã—3/2
a_{3}Â =Â 8
Therefore, the missing terms are 13/2 and 8 respectively.
(iv) For the given A.P.,
aÂ = âˆ’4 and
a_{6}Â = 6
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Therefore, putting the values here,
a_{6}Â = a + (6 âˆ’ 1) d
6 = âˆ’ 4 + 5d
10 = 5d
dÂ = 2
a_{2}Â =Â aÂ +Â dÂ = âˆ’ 4 + 2 = âˆ’2
a_{3}Â =Â aÂ + 2dÂ = âˆ’ 4 + 2 (2) = 0
a_{4}Â =Â aÂ + 3dÂ = âˆ’ 4 + 3 (2) = 2
a_{5}Â =Â aÂ + 4dÂ = âˆ’ 4 + 4 (2) = 4
Therefore, the missing terms are âˆ’2, 0, 2, and 4 respectively.
(v) For the given A.P.,
a_{2}Â = 38
a_{6}Â = âˆ’22
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Therefore, putting the values here,
a_{2}Â =Â aÂ + (2 âˆ’ 1)Â d
38 =Â aÂ +Â dÂ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.Â (i)
a_{6}Â =Â aÂ + (6 âˆ’ 1)Â d
âˆ’22 =Â aÂ + 5dÂ â€¦â€¦â€¦â€¦â€¦â€¦â€¦.Â (ii)
On subtracting equationÂ (i)Â fromÂ (ii), we get
âˆ’ 22 âˆ’ 38 = 4d
âˆ’60 = 4d
dÂ = âˆ’15
aÂ =Â a_{2}Â âˆ’Â dÂ = 38 âˆ’ (âˆ’15) = 53
a_{3}Â =Â aÂ + 2dÂ = 53 + 2 (âˆ’15) = 23
a_{4}Â =Â aÂ + 3dÂ = 53 + 3 (âˆ’15) = 8
a_{5}Â =Â aÂ + 4dÂ = 53 + 4 (âˆ’15) = âˆ’7
Therefore, the missing terms are 53, 23, 8, and âˆ’7 respectively.
4. Which term of the A.P. 3, 8, 13, 18, â€¦ is 78?
Solutions:
Given the A.P. series as3, 8, 13, 18, â€¦
Thus,
First term, aÂ = 3
Common difference, dÂ =Â a_{2}Â âˆ’Â a_{1}Â = 8 âˆ’ 3 = 5
LetÂ the n^{th}Â term of given A.P. be 78. Now as we know,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Therefore,
78 = 3 + (nÂ âˆ’ 1) 5
75 = (nÂ âˆ’ 1) 5
(nÂ âˆ’ 1) = 15
nÂ = 16
Hence, 16^{th}Â term of this A.P. is 78.
5. Find the number of terms in each of the following A.P.
(i) 7, 13, 19, â€¦, 205
(ii) 18,, 13,â€¦., -47
Solutions:
(i) Given, 7, 13, 19, â€¦, 205 is the A.P
Therefore,
First term, aÂ = 7
Common difference, dÂ =Â a_{2}Â âˆ’Â a_{1}Â = 13 âˆ’ 7 = 6
Let there areÂ nÂ terms in this A.P.
a_{n}Â = 205
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Therefore, 205 = 7 + (nÂ âˆ’ 1) 6
198 = (nÂ âˆ’ 1) 6
33 = (nÂ âˆ’ 1)
nÂ = 34
Therefore, this given series has 34 terms in it.
(ii) Given, 18,13,â€¦., -47 is the A.P.
First term, aÂ = 18
Common difference, d = a_{2}-a_{1 }= 15/2â€“ 18
d=31-36/2 = â€“ 5/2
Let there are n terms in this A.P.
a_{n}Â = 205
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
-47 = 18Â + (nÂ â€“ 1) (-5/2)
-47 â€“ 18 = (nÂ â€“ 1) (-5/2)
-65 = (nÂ â€“ 1)(-5/2)
(nÂ â€“ 1) = -130/-5
(nÂ â€“ 1) =Â 26
nÂ = 27
Therefore, this given A.P. has 27 terms in it.
6.Â Check whether -150 is a term of the A.P. 11, 8, 5, 2, â€¦
Solution:
For the given series, A.P. 11, 8, 5, 2, â€¦
First term, aÂ = 11
Common difference, dÂ =Â a_{2}Â âˆ’Â a_{1}Â = 8 âˆ’ 11 = âˆ’3
Let âˆ’150 be theÂ n^{th}Â term of this A.P.
As we know, for an A.P.,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
-150 = 11Â + (nÂ â€“ 1)(-3)
-150 = 11 â€“ 3nÂ + 3
-164 = -3n
nÂ = 164/3
Clearly,Â nÂ is not an integer but a fraction.
Therefore, â€“ 150 is not a term of this A.P.
7. Find the 31^{st}Â term of an A.P. whose 11^{th}Â term is 38 and the 16^{th}Â term is 73.
Solution:
Given that,
11^{th} term, a_{11}Â = 38
and 16^{th} term, a_{16}Â = 73
We know that,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
a_{11}Â =Â aÂ + (11 âˆ’ 1)Â d
38 =Â aÂ + 10dÂ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.Â (i)
In the same way,
a_{16}Â =Â aÂ + (16 âˆ’ 1)Â d
73 =Â aÂ + 15dÂ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦Â (ii)
On subtracting equationÂ (i)Â fromÂ (ii), we get
35 = 5d
dÂ = 7
From equationÂ (i), we can write,
38 =Â aÂ + 10 Ã— (7)
38 âˆ’ 70 =Â a
aÂ = âˆ’32
a_{31}Â =Â aÂ + (31 âˆ’ 1)Â d
= âˆ’ 32 + 30 (7)
= âˆ’ 32 + 210
= 178
Hence, 31^{st}Â term is 178.
8. An A.P. consists of 50 terms of which 3^{rd}Â term is 12 and the last term is 106. Find the 29^{th}Â term.
Solution:Given that,
3^{rd} term, a_{3}Â = 12
50^{th} term, a_{50}Â = 106
We know that,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
a_{3}Â =Â aÂ + (3 âˆ’ 1)Â d
12 =Â aÂ + 2dÂ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.Â (i)
In the same way,
a_{50Â }=Â aÂ + (50 âˆ’ 1)Â d
106 =Â aÂ + 49dÂ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.Â (ii)
On subtracting equationÂ (i)Â fromÂ (ii), we get
94 = 47d
dÂ = 2 = common difference
From equationÂ (i), we can write now,
12 =Â aÂ + 2 (2)
aÂ = 12 âˆ’ 4 = 8
a_{29}Â =Â aÂ + (29 âˆ’ 1)Â d
a_{29}Â = 8 + (28)2
a_{29}Â = 8 + 56 = 64
Therefore, 29^{th}Â term is 64.
9. If the 3^{rd}Â and the 9^{th}Â terms of an A.P. are 4 and âˆ’ 8 respectively. Which term of this A.P. is zero.
Solution:
Given that,
3^{rd} term, a_{3}Â = 4
and 9^{th} term, a_{9}Â = âˆ’8
We know that,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Therefore,
a_{3}Â =Â aÂ + (3 âˆ’ 1)Â d
4 =Â aÂ + 2dÂ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦Â (i)
a_{9}Â =Â aÂ + (9 âˆ’ 1)Â d
âˆ’8 =Â aÂ + 8dÂ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦Â (ii)
On subtracting equationÂ (i)Â fromÂ (ii), we will get here,
âˆ’12 = 6d
dÂ = âˆ’2
From equationÂ (i), we can write,
4 =Â aÂ + 2 (âˆ’2)
4 =Â aÂ âˆ’ 4
aÂ = 8
LetÂ n^{th}Â term of this A.P. be zero.
a_{nÂ }=Â aÂ + (nÂ âˆ’ 1)Â d
0 = 8 + (nÂ âˆ’ 1) (âˆ’2)
0 = 8 âˆ’ 2nÂ + 2
2nÂ = 10
nÂ = 5
Hence, 5^{th}Â term of this A.P. is 0.
10. If 17^{th}Â term of an A.P. exceeds its 10^{th}Â term by 7. Find the common difference.
Solution: We know that, for an A.P series;
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
a_{17}Â =Â aÂ + (17 âˆ’ 1)Â d
a_{17}Â =Â aÂ + 16d
In the same way,
a_{10}Â =Â aÂ + 9d
As it is given in the question,
a_{17}Â âˆ’Â a_{10}Â = 7
Therefore,
(aÂ + 16d) âˆ’ (aÂ + 9d) = 7
7dÂ = 7
dÂ = 1
Therefore, the common difference is 1.
11.Â Which term of the A.P. 3, 15, 27, 39, â€¦ will be 132 more than its 54^{th}Â term?
Solution: Given A.P. is 3, 15, 27, 39, â€¦
first term, aÂ = 3
common difference, dÂ =Â a_{2}Â âˆ’Â a_{1}Â = 15 âˆ’ 3 = 12
We know that,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Therefore,
a_{54}Â =Â aÂ + (54 âˆ’ 1)Â d
â‡’ 3 + (53) (12)
â‡’ 3 + 636 = 639
a_{54 }= 639
We have to find the term of this A.P. which is 132 more than a_{54, }i.e.771.
LetÂ n^{th}Â term be 771.
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
771 = 3 + (nÂ âˆ’ 1) 12
768 = (nÂ âˆ’ 1) 12
(nÂ âˆ’ 1) = 64
nÂ = 65
Therefore, 65^{th}Â term was 132 more than 54^{th}Â term.
Or another method is;
LetÂ n^{th}Â term be 132 more than 54^{th}Â term.
nÂ = 54Â + 132/2
= 54Â + 11 =Â 65^{th}Â term
12. Two APs have the same common difference. The difference between their 100^{th}Â term is 100, what is the difference between their 1000^{th}Â terms?
Solution: Let,
The first term of two APs beÂ a_{1}Â andÂ a_{2}Â respectively
And the common difference of these APs beÂ d.
For the first A.P.,we know,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Therefore,
a_{100}Â =Â a_{1}Â + (100 âˆ’ 1)Â d
=Â a_{1}Â + 99d
a_{1000}Â =Â a_{1}Â + (1000 âˆ’ 1)Â d
a_{1000}Â =Â a_{1}Â + 999d
For second A.P., we know,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Therefore,
a_{100}Â =Â a_{2}Â + (100 âˆ’ 1)Â d
=Â a_{2}Â + 99d
a_{1000}Â =Â a_{2}Â + (1000 âˆ’ 1)Â d
=Â a_{2}Â + 999d
Given that, difference between 100^{th}Â term of the two APs = 100
Therefore, (a_{1}Â + 99d) âˆ’ (a_{2}Â + 99d) = 100
a_{1}Â âˆ’Â a_{2}Â = 100 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.Â (i)
Difference between 1000^{th}Â terms of the two APs
(a_{1}Â + 999d) âˆ’ (a_{2}Â + 999d) =Â a_{1}Â âˆ’Â a_{2}
From equationÂ (i),
This difference,Â a_{1}Â âˆ’Â a_{2Â }= 100
Hence, the difference between 1000^{th}Â terms of the two A.P. will be 100.
13. How many three digit numbers are divisible by 7?
Solution: First three-digit number that is divisible by 7 are;
First number = 105
Second number = 105 + 7 = 112
Third number = 112+7 =119
Therefore, 105, 112, 119, â€¦
All are three digit numbers are divisible by 7 and thus, all these are terms of an A.P. having first term as 105 and common difference as 7.
As we know, the largest possible three-digit number is 999.
When we divide 999 by 7, the remainder will be 5.
Therefore, 999 âˆ’ 5 = 994 is the maximum possible three-digit number that is divisible by 7.
Now the series is as follows.
105, 112, 119, â€¦, 994
Let 994 be theÂ nth term of this A.P.
first term, aÂ = 105
common difference, dÂ = 7
a_{n}Â = 994
nÂ = ?
As we know,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
994 = 105 + (nÂ âˆ’ 1) 7
889 = (nÂ âˆ’ 1) 7
(nÂ âˆ’ 1) = 127
nÂ = 128
Therefore, 128 three-digit numbers are divisible by 7.
14. How many multiples of 4 lie between 10 and 250?
Solution: The first multiple of 4 that is greater than 10 is 12.
Next multiple will be 16.
Therefore, the series formed as;
12, 16, 20, 24, â€¦
All these are divisible by 4 and thus, all these are terms of an A.P. with first term as 12 and common difference as 4.
When we divide 250 by 4, the remainder will be 2. Therefore, 250 âˆ’ 2 = 248 is divisible by 4.
The series is as follows, now;
12, 16, 20, 24, â€¦, 248
Let 248 be theÂ n^{th}Â term of this A.P.
first term, aÂ = 12
common difference, dÂ = 4
a_{n}Â =Â 248
As we know,
a_{n}Â =Â aÂ + (nÂ â€“ 1)Â d
248 = 12Â + (nÂ â€“ 1) Ã— 4
236/4 =Â nÂ â€“ 1
59 Â =Â nÂ â€“ 1
nÂ = 60
Therefore, there are 60 multiples of 4 between 10 and 250.
15. For what value ofÂ n, are theÂ n^{th}Â terms of two APs 63, 65, 67, and 3, 10, 17, â€¦ equal?
Solution: Given two Aps as; 63, 65, 67,â€¦ and 3, 10, 17,â€¦.
Taking first AP,
63, 65, 67, â€¦
First term, aÂ = 63
Common difference, dÂ =Â a_{2}Â âˆ’Â a_{1}Â = 65 âˆ’ 63 = 2
We know, n^{th}Â term of this A.P. =Â a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
a_{n}= 63 + (nÂ âˆ’ 1) 2 = 63 + 2nÂ âˆ’ 2
a_{n}Â = 61 + 2nÂ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.Â (i)
Taking second AP,
3, 10, 17, â€¦
First term, aÂ = 3
Common difference, dÂ =Â a_{2}Â âˆ’Â a_{1}Â = 10 âˆ’ 3 = 7
We know that,
n^{th}Â term of this A.P. = 3 + (nÂ âˆ’ 1) 7
a_{n}Â = 3 + 7nÂ âˆ’ 7
a_{n}Â = 7nÂ âˆ’ 4 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..Â (ii)
Given,Â n^{th}Â term of these A.P.s are equal to each other.
Equating both these equations, we get,
61 + 2nÂ = 7nÂ âˆ’ 4
61 + 4 = 5n
5nÂ = 65
nÂ = 13
Therefore, 13^{th}Â terms of both these A.P.s are equal to each other.
16. Determine the A.P. whose third term is 16 and the 7^{th}Â term exceeds the 5^{th}Â term by 12.
Solutions: Given,
Third term, a_{3}Â = 16
As we know,
aÂ + (3 âˆ’ 1)Â dÂ = 16
aÂ + 2dÂ = 16 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.Â (i)
It is given that, 7^{th}Â term exceeds the 5^{th}Â term by 12.
a_{7}Â âˆ’Â a_{5}Â = 12
[a+ (7 âˆ’ 1)Â d] âˆ’ [aÂ + (5 âˆ’ 1)Â d]= 12
(aÂ + 6d) âˆ’ (aÂ + 4d) = 12
2dÂ = 12
dÂ = 6
From equationÂ (i), we get,
aÂ + 2 (6) = 16
aÂ + 12 = 16
aÂ = 4
Therefore, A.P. will be 4, 10, 16, 22, â€¦
17. Find the 20^{th}Â term from the last term of the A.P. 3, 8, 13, â€¦, 253.
Solutions: Given A.P. is 3, 8, 13, â€¦, 253
Common difference,d = 5.
Therefore, we can write the given AP in reverse order as;
253, 248, 243, â€¦, 13, 8, 5
Now for the new AP,
first term, aÂ = 253
and common difference, dÂ = 248 âˆ’ 253 = âˆ’5
nÂ = 20
Therefore, using nth term formula, we get,
a_{20}Â =Â aÂ + (20 âˆ’ 1)Â d
a_{20}Â = 253 + (19) (âˆ’5)
a_{20}Â = 253 âˆ’ 95
aÂ = 158
Therefore, 20^{th}Â term from the last term of the AP 3, 8, 13, â€¦, 253. is 158.
18. The sum of 4^{th}Â and 8^{th}Â terms of an A.P. is 24 and the sum of the 6^{th}Â and 10^{th}Â terms is 44. Find the first three terms of the A.P.
Solutions: We know that, the nth term of the AP is;
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
a_{4}Â =Â aÂ + (4 âˆ’ 1)Â d
a_{4}Â =Â aÂ + 3d
In the same way, we can write,
a_{8}Â =Â aÂ + 7d
a_{6}Â =Â aÂ + 5d
a_{10}Â =Â aÂ + 9d
Given that,
a_{4}Â +Â a_{8}Â = 24
aÂ + 3dÂ +Â aÂ + 7dÂ = 24
2aÂ + 10dÂ = 24
aÂ + 5dÂ = 12 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦Â (i)
a_{6}Â +Â a_{10}Â = 44
aÂ + 5dÂ +Â aÂ + 9dÂ = 44
2aÂ + 14dÂ = 44
aÂ + 7dÂ = 22 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..Â (ii)
On subtracting equationÂ (i)Â fromÂ (ii), we get,
2dÂ = 22 âˆ’ 12
2dÂ = 10
dÂ = 5
From equationÂ (i), we get,
aÂ + 5dÂ = 12
aÂ + 5 (5) = 12
aÂ + 25 = 12
aÂ = âˆ’13
a_{2}Â =Â aÂ +Â dÂ = âˆ’ 13 + 5 = âˆ’8
a_{3}Â =Â a_{2}Â +Â dÂ = âˆ’ 8 + 5 = âˆ’3
Therefore, the first three terms of this A.P. are âˆ’13, âˆ’8, and âˆ’3.
19. Subba Rao started work in 1995 at an annual salary of Rs 5000 and received an increment of Rs 200 each year. In which year did his income reach Rs 7000?
Solution: It can be seen from the given question, that the incomes of Subba Rao increases every year by Rs.200 and hence, forms an AP.
Therefore, after 1995, the salaries of each year are;
5000, 5200, 5400, â€¦
Here,Â first term, aÂ = 5000
and common difference, dÂ = 200
Let afterÂ n^{th}Â year, his salary be Rs 7000.
Therefore, by the nth term formula of AP,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
7000 = 5000 + (nÂ âˆ’ 1) 200
200(nÂ âˆ’ 1) = 2000
(nÂ âˆ’ 1) = 10
nÂ = 11
Therefore, in 11th year, his salary will be Rs 7000.
20. Ramkali saved Rs 5 in the first week of a year and then increased her weekly saving by Rs 1.75. If in theÂ n^{th}Â week, her week, her weekly savings become Rs 20.75, findÂ n.
Solution: Given that, Ramkali saved Rs.5 in first week and then started saving each week by Rs.1.75.
Hence,
First term, aÂ = 5
and common difference, dÂ = 1.75
Also given,
a_{nÂ }= 20.75
Find, nÂ = ?
As we know, by the nth term formula,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)Â d
Therefore,
20.75 = 5Â + (nÂ â€“ 1) Ã— 1.75
15.75 = (nÂ â€“ 1) Ã— 1.75
(nÂ â€“ 1) = 15.75/1.75 = 1575/175
= 63/7 = 9
nÂ â€“ 1 = 9
nÂ = 10
Hence,Â nÂ is 10.
Class 10 Maths Chapter 5 Exercise 5.3 Page: 112
1. Find the sum of the following APs.
(i) 2, 7, 12 ,â€¦., to 10 terms.
(ii) âˆ’ 37, âˆ’ 33, âˆ’ 29 ,â€¦, to 12 terms
(iii) 0.6, 1.7, 2.8 ,â€¦â€¦.., to 100 terms
(iv) 1/15, 1/12, 1/10, â€¦â€¦ , to 11 terms
Solutions:
(i) Given, 2, 7, 12 ,â€¦, to 10 terms
For this A.P.,
first term, aÂ = 2
And common difference, dÂ =Â a_{2}Â âˆ’Â a_{1}Â = 7 âˆ’ 2 = 5
nÂ = 10
We know that, the formula for sum of nth term in AP series is,
S_{n}Â =Â n/2Â [2aÂ +Â (nÂ â€“ 1)Â d]
S_{10}Â = 10/2Â [2(2)Â +Â (10Â â€“ 1) Ã— 5]
= 5[4Â +Â (9) Ã—Â (5)]
= 5 Ã— 49 = 245
(ii) Given, âˆ’37, âˆ’33, âˆ’29 ,â€¦, to 12 terms
For this A.P.,
first term, aÂ = âˆ’37
And common difference, dÂ =Â a_{2}Â âˆ’Â a_{1}Â = (âˆ’33) âˆ’ (âˆ’37)
= âˆ’ 33 + 37 = 4
nÂ = 12
We know that, the formula for sum of nth term in AP series is,
S_{n}Â =Â n/2Â [2aÂ +Â (nÂ â€“ 1)Â d]
S_{12}Â = 12/2Â [2(-37)Â +Â (12 â€“ 1) Ã— 4]
= 6[-74Â +Â 11 Ã—Â 4]
= 6[-74Â +Â 44]
= 6(-30) = -180
(iii) Given, 0.6, 1.7, 2.8 ,â€¦, to 100 terms
For this A.P.,
first term, aÂ = 0.6
and
Common difference, dÂ =Â a_{2}Â âˆ’Â a_{1}Â = 1.7 âˆ’ 0.6 = 1.1
nÂ = 100
We know that, the formula for sum of nth term in AP series is,
S_{n}Â =Â n/2Â [2aÂ +Â (nÂ â€“ 1)Â d]
S_{12}Â = 50/2Â [1.2 +Â (99) Ã— 1.1]
= 50[1.2 +Â 108.9]
= 50[110.1]
= 5505
2. Find the sums given below:
(i) 7 + 10(1/2)Â + 14Â + â€¦â€¦â€¦â€¦â€¦â€¦ +84
(ii) 34 + 32 + 30 + â€¦â€¦â€¦.. + 10
(iii) âˆ’ 5 + (âˆ’ 8) + (âˆ’ 11) + â€¦â€¦â€¦â€¦ + (âˆ’ 230)
Solutions:
(ii) Given, 34 + 32 + 30 + â€¦â€¦â€¦.. + 10
For this A.P.,
first term, aÂ = 34
common difference, dÂ =Â a_{2}Â âˆ’Â a_{1}Â = 32 âˆ’ 34 = âˆ’2
nth term, a_{n} = 10
Let 10 be theÂ n^{th}Â term of this A.P., therefore,
a_{n} =Â aÂ + (nÂ âˆ’ 1)Â d
10 = 34 + (nÂ âˆ’ 1) (âˆ’2)
âˆ’24 = (nÂ âˆ’ 1) (âˆ’2)
12 =Â nÂ âˆ’ 1
nÂ = 13
We know that, sum of nth term is;
S_{n}Â =Â n/2 (aÂ +Â l)
= 13/2 (34Â + 10)
= (13Ã—44/2) = 13 Ã— 22
= 286
(iii)Â Given, (âˆ’5) + (âˆ’8) + (âˆ’11) + â€¦â€¦â€¦â€¦ + (âˆ’230)
For this A.P.,
First term, aÂ = âˆ’5
nth term,Â a_{n} = âˆ’230
Common difference, dÂ =Â a_{2}Â âˆ’Â a_{1}Â = (âˆ’8) âˆ’ (âˆ’5)
â‡’ d = âˆ’ 8 + 5 = âˆ’3
Let âˆ’230 be theÂ n^{th}Â term of this A.P., and by the nth term formula we know,
a_{n} =Â aÂ + (nÂ âˆ’ 1)d
âˆ’230 = âˆ’ 5 + (nÂ âˆ’ 1) (âˆ’3)
âˆ’225 = (nÂ âˆ’ 1) (âˆ’3)
(nÂ âˆ’ 1) = 75
nÂ = 76
And, Sum of nth term,
S_{n}Â =Â n/2 (aÂ +Â l)
= 76/2Â [(-5) +Â (-230)]
= 38(-235)
= -8930
3. In an AP
(i) GivenÂ aÂ = 5,Â dÂ = 3,Â a_{n}Â = 50, findÂ nÂ andÂ S_{n}.
(ii) GivenÂ aÂ = 7,Â a_{13}Â = 35, findÂ dÂ andÂ S_{13}.
(iii) GivenÂ a_{12}Â = 37,Â dÂ = 3, findÂ aÂ andÂ S_{12}.
(iv) GivenÂ a_{3}Â = 15,Â S_{10}Â = 125, findÂ dÂ andÂ a_{10}.
(v) GivenÂ dÂ = 5,Â S_{9}Â = 75, findÂ aÂ andÂ a_{9}.
(vi) GivenÂ aÂ = 2,Â dÂ = 8,Â S_{n}Â = 90, findÂ nÂ andÂ a_{n}.
(vii) GivenÂ aÂ = 8,Â a_{n}Â = 62,Â S_{n}Â = 210, findÂ nÂ andÂ d.
(viii) GivenÂ a_{n}Â = 4,Â dÂ = 2,Â S_{n}Â = âˆ’ 14, findÂ nÂ andÂ a.
(ix) GivenÂ aÂ = 3,Â nÂ = 8,Â SÂ = 192, findÂ d.
(x) GivenÂ lÂ = 28,Â SÂ = 144 and there are total 9 terms. FindÂ a.
Solutions:
(i) Given that,Â aÂ = 5,Â dÂ = 3,Â a_{n}Â = 50
AsÂ we know, from the formula of the nth term in an AP,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)d,
Therefore, putting the given values, we get,
â‡’ 50 = 5Â + (nÂ â€“ 1) Ã— 3
â‡’ 3(nÂ â€“ 1) = 45
â‡’Â nÂ â€“ 1 = 15
â‡’Â nÂ = 16
Now, sum of nth term,
S_{n}Â =Â n/2 (aÂ +Â a_{n})
S_{n}Â = 16/2 (5Â + 50) = 440
(ii)Â Given that,Â aÂ = 7,Â a_{13}Â = 35
AsÂ we know, from the formula of the nth term in an AP,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)d,
Therefore, putting the given values, we get,
â‡’ 35 = 7Â + (13 â€“ 1)d
â‡’ 12dÂ = 28
â‡’Â dÂ = 28/12 = 2.33
Now,Â S_{n}Â =Â n/2 (aÂ +Â a_{n})
S_{13}Â =Â 13/2 (7Â + 35) = 273
(iii)Given that,Â a_{12}Â = 37,Â dÂ = 3
AsÂ we know, from the formula of the nth term in an AP,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)d,
Therefore, putting the given values, we get,
â‡’Â a_{12}Â =Â aÂ + (12 âˆ’ 1)3
â‡’ 37 =Â aÂ + 33
â‡’Â aÂ = 4
Now, sum of nth term,
S_{n}Â =Â n/2 (aÂ +Â a_{n})
S_{n}Â =Â 12/2 (4 + 37)
= 246
(iv) Given that,Â a_{3}Â = 15,Â S_{10}Â = 125
AsÂ we know, from the formula of the nth term in an AP,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)d,
Therefore, putting the given values, we get,
a_{3}Â =Â aÂ + (3 âˆ’ 1)d
15 =Â aÂ + 2dÂ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..Â (i)
Sum of the nth term,
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
S_{10}Â =Â 10/2Â [2aÂ + (10Â â€“ 1)d]
125 = 5(2aÂ + 9d)
25 = 2aÂ + 9dÂ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..Â (ii)
On multiplying equationÂ (i)Â byÂ (ii), we will get;
30 = 2aÂ + 4dÂ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.Â (iii)
By subtracting equationÂ (iii)Â fromÂ (ii), we get,
âˆ’5 = 5d
dÂ = âˆ’1
From equationÂ (i),
15 =Â aÂ + 2(âˆ’1)
15 =Â aÂ âˆ’ 2
aÂ = 17 = First term
a_{10}Â =Â aÂ + (10 âˆ’ 1)d
a_{10}Â = 17 + (9) (âˆ’1)
a_{10}Â = 17 âˆ’ 9 = 8
(v) Given that,Â dÂ = 5,Â S_{9}Â = 75
As, sum of nth terms in AP is,
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
Therfore, the sum of first nine terms are;
S_{9}Â = 9/2Â [2aÂ + (9 â€“ 1)5]
25 = 3(aÂ + 20)
25 = 3aÂ + 60
3aÂ = 25 âˆ’ 60
aÂ = -35/3
As we know, the nth term can be written as;
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)d
a_{9}Â =Â aÂ + (9 âˆ’ 1) (5)
= -35/3Â + 8(5)
= -35/3Â + 40
= (35+120/3) = 85/3
(vi) Given that,Â aÂ = 2,Â dÂ = 8,Â S_{n}Â = 90
As, sum of nth term in an AP is,
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
90 =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
â‡’ 180 =Â n(4Â + 8nÂ â€“ 8) =Â n(8nÂ â€“ 4) = 8n^{2}Â â€“ 4n
â‡’ 8n^{2}Â â€“ 4n â€“Â 180 = 0
â‡’ 2n^{2}Â â€“Â nÂ â€“ 45 = 0
â‡’ 2n^{2}Â â€“Â 10nÂ + 9nÂ â€“ 45 = 0
â‡’ 2n(nÂ -5)Â + 9(nÂ â€“ 5) = 0
â‡’ (2nÂ â€“ 9)(2nÂ + 9) = 0
So,Â nÂ = 5 (as it is positive integer)
âˆ´Â a_{5}_{Â }= 8Â + 5 Ã— 4 = 34
(vii) Given that,Â aÂ = 8,Â a_{n}Â = 62,Â S_{n}Â = 210
As, sum of nth term in an AP is,
S_{n}Â =Â n/2 (aÂ +Â a_{n})
210 =Â n/2 (8 + 62)
â‡’ 35nÂ = 210
â‡’Â nÂ = 210/35 = 6
Now, 62 = 8Â + 5d
â‡’ 5dÂ = 62 â€“ 8 = 54
â‡’Â dÂ = 54/5 = 10.8
(viii) Given that,Â nth term, a_{n}Â = 4,Â common difference, dÂ = 2,Â sum of nth term, S_{n}Â = âˆ’14.
AsÂ we know, from the formula of the nth term in an AP,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)d,
Therefore, putting the given values, we get,
4 =Â aÂ + (nÂ âˆ’ 1)2
4 =Â aÂ + 2nÂ âˆ’ 2
aÂ + 2nÂ = 6
aÂ = 6 âˆ’ 2nÂ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.Â (i)
As we know, the sum of nth term is;
S_{n}Â =Â n/2 (aÂ +Â a_{n})
-14 =Â n/2 (aÂ +Â 4)
âˆ’28 =Â nÂ (aÂ + 4)
âˆ’28 =Â nÂ (6 âˆ’ 2nÂ + 4) {From equationÂ (i)}
âˆ’28 =Â nÂ (âˆ’ 2nÂ + 10)
âˆ’28 = âˆ’ 2n^{2}Â + 10n
2n^{2}Â âˆ’ 10nÂ âˆ’ 28 = 0
n^{2}Â âˆ’ 5nÂ âˆ’14 = 0
n^{2}Â âˆ’ 7n +Â 2nÂ âˆ’ 14 = 0
nÂ (nÂ âˆ’ 7) + 2(nÂ âˆ’ 7) = 0
(nÂ âˆ’ 7) (nÂ + 2) = 0
EitherÂ nÂ âˆ’ 7 = 0 orÂ nÂ + 2 = 0
nÂ = 7 orÂ nÂ = âˆ’2
However,Â nÂ can neither be negative nor fractional.
Therefore,Â nÂ = 7
From equationÂ (i), we get
aÂ = 6 âˆ’ 2n
aÂ = 6 âˆ’ 2(7)
= 6 âˆ’ 14
= âˆ’8
(ix) Given that,Â first term, aÂ = 3,
Number of terms,Â nÂ = 8
AndÂ sum of nth term, SÂ = 192
As we know,
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
192 = 8/2Â [2 Ã— 3Â + (8Â â€“ 1)d]
192 = 4 [6 + 7d]
48 = 6 + 7d
42 = 7d
d =Â 6
(x) Given that,Â lÂ = 28, SÂ = 144 and there are total of 9 terms.
Sum of nth term formula,
S_{n}Â =Â n/2 (aÂ +Â l)
144 = 9/2 (aÂ + 28)
(16) Ã— (2) =Â aÂ + 28
32 =Â aÂ + 28
aÂ = 4
4. How many terms of the AP. 9, 17, 25 â€¦ must be taken to give a sum of 636?
Solutions:
Let there beÂ nÂ terms of the AP. 9, 17, 25 â€¦
For this A.P.,
First term, aÂ = 9
Common difference, dÂ =Â a_{2}Â âˆ’Â a_{1}Â = 17 âˆ’ 9 = 8
As, the sum of nth terms, is;
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
636 =Â n/2Â [2 Ã—Â aÂ + (8Â â€“ 1) Ã— 8]
636 =Â n/2Â [18 + (nâ€“ 1) Ã— 8]
636 =Â nÂ [9 + 4nÂ âˆ’ 4]
636 =Â nÂ (4nÂ + 5)
4n^{2}Â + 5nÂ âˆ’ 636 = 0
4n^{2}Â + 53nÂ âˆ’ 48nÂ âˆ’ 636 = 0
nÂ (4nÂ + 53) âˆ’ 12 (4nÂ + 53) = 0
(4nÂ + 53) (nÂ âˆ’ 12) = 0
Either 4nÂ + 53 = 0 orÂ nÂ âˆ’ 12 = 0
nÂ = (-53/4) orÂ nÂ = 12
nÂ cannot be negative or fraction, therefore,Â nÂ = 12 only.
5. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
Solution: Given that,
first term, aÂ = 5
last term, lÂ = 45
Sum of the AP, S_{n}Â = 400
As we know, the sum of AP formula is;
S_{n}Â =Â n/2 (aÂ +Â l)
400 =Â n/2 (5Â + 45)
400 =Â n/2 (50)
Number of terms, nÂ = 16
As we know, the last term of AP series can be written as;
l = a +Â (nÂ âˆ’ 1)Â d
45 = 5 + (16 âˆ’ 1)Â d
40 = 15d
Common difference, dÂ = 40/15 = 8/3
6. The first and the last term of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?
Solution: Given that,
First term, aÂ = 17
Last term, lÂ = 350
Common difference, dÂ = 9
Let there beÂ nÂ terms in the A.P., thus the formula for last term can be written as;
l = a +Â (nÂ âˆ’ 1)Â d
350 = 17 + (nÂ âˆ’ 1)9
333 = (nÂ âˆ’ 1)9
(nÂ âˆ’ 1) = 37
nÂ = 38
S_{n}Â =Â n/2 (aÂ +Â l)
S_{38}Â =Â 13/2 (17Â + 350)
= 19 Ã— 367
= 6973
Thus, this A.P. contains 38 terms and the sum of the terms of this A.P. is 6973.
7. Find the sum of first 22 terms of an AP in whichÂ dÂ = 7 and 22^{nd}Â term is 149.
Solution:Given,
Common difference, dÂ = 7
22^{nd} term, a_{22}Â = 149
Sum of first 22 term, S_{22}Â = ?
By the formula of nth term,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)d
a_{22}Â =Â aÂ + (22 âˆ’ 1)d
149 =Â aÂ + 21 Ã— 7
149 =Â aÂ + 147
aÂ = 2 = First term
Sum of nth term,
S_{n}Â =Â n/2 (aÂ +Â a_{n})
= 22/2 (2Â +Â 149)
= 11 Ã— 151
= 1661
8. Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.
Solution: Given that,
Second term, a_{2}Â = 14
Third term, a_{3}Â = 18
Common difference, dÂ =Â a_{3}Â âˆ’Â a_{2}Â = 18 âˆ’ 14 = 4
a_{2}Â =Â aÂ +Â d
14 =Â aÂ + 4
aÂ = 10 = First term
Sum of nth term;
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
S_{51}Â = 51/2Â [2 Ã— 10Â + (51 â€“ 1) Ã— 4]
= 51/2Â [2 + (20) Ã— 4]
= 51Ã—220/2
= 51 Ã— 110
= 5610
9. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of firstÂ nÂ terms.
Solution: Given that,
S_{7}Â = 49
S_{17}Â = 289
We know, Sum of nth term;
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
Therefore,
S_{7} =Â 7/2Â [2aÂ + (nÂ â€“ 1)d]
S_{7}Â = 7/2Â [2aÂ + (7 â€“ 1)d]
49 = 7/2Â [2aÂ +Â 16d]
7 = (aÂ + 3d)
aÂ + 3dÂ = 7 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.Â (i)
In the same way,
S_{17}Â = 17/2Â [2aÂ + (17 â€“ 1)d]
289 = 17/2 (2aÂ + 16d)
17 = (aÂ + 8d)
aÂ + 8dÂ = 17 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.Â (ii)
Subtracting equationÂ (i)Â from equationÂ (ii),
5dÂ = 10
dÂ = 2
From equationÂ (i), we can write it as;
aÂ + 3(2) = 7
a +Â 6 = 7
a =Â 1
Hence,
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
=Â n/2Â [2(1)Â + (nÂ â€“ 1)Â Ã— 2]
=Â n/2 (2Â + 2nÂ â€“ 2)
=Â n/2 (2n)
=Â n^{2}
^{}10. Show thatÂ a_{1},Â a_{2Â }â€¦ ,Â a_{n}Â , â€¦ form an AP whereÂ a_{n}Â is defined as below
(i)Â a_{n}Â = 3 + 4n
(ii)Â a_{n}Â = 9 âˆ’ 5n
Also find the sum of the first 15 terms in each case.
Solutions:
(i)Â a_{n}Â = 3 + 4n
a_{1}Â = 3 + 4(1) = 7
a_{2}Â = 3 + 4(2) = 3 + 8 = 11
a_{3}Â = 3 + 4(3) = 3 + 12 = 15
a_{4}Â = 3 + 4(4) = 3 + 16 = 19
We can see here, the common difference between the terms are;
a_{2}Â âˆ’Â a_{1}Â = 11 âˆ’ 7 = 4
a_{3}Â âˆ’Â a_{2}Â = 15 âˆ’ 11 = 4
a_{4}Â âˆ’Â a_{3}Â = 19 âˆ’ 15 = 4
Hence,Â a_{k}_{Â + 1}Â âˆ’Â a_{k}Â is the same value every time. Therefore, this is an AP with common difference as 4 and first term as 7.
Now, we know, the sum of nth term is;
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
S_{15Â }= 15/2Â [2(7)Â + (15Â â€“ 1)Â Ã— 4]
= 15/2Â [(14)Â + 56]
= 15/2 (70)
= 15 Ã— 35
= 525
(ii)Â a_{n}Â = 9 âˆ’ 5n
a_{1}Â = 9 âˆ’ 5 Ã— 1 = 9 âˆ’ 5 = 4
a_{2}Â = 9 âˆ’ 5 Ã— 2 = 9 âˆ’ 10 = âˆ’1
a_{3}Â = 9 âˆ’ 5 Ã— 3 = 9 âˆ’ 15 = âˆ’6
a_{4}Â = 9 âˆ’ 5 Ã— 4 = 9 âˆ’ 20 = âˆ’11
We can see here, the common difference between the terms are;
a_{2}Â âˆ’Â a_{1}Â = âˆ’ 1 âˆ’ 4 = âˆ’5
a_{3}Â âˆ’Â a_{2}Â = âˆ’ 6 âˆ’ (âˆ’1) = âˆ’5
a_{4}Â âˆ’Â a_{3}Â = âˆ’ 11 âˆ’ (âˆ’6) = âˆ’5
Hence, a_{k}_{Â + 1}Â âˆ’Â a_{k}Â is same every time. Therefore, this is an A.P. with common difference as âˆ’5 and first term as 4.
Now, we know, the sum of nth term is;
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
S_{15Â }= 15/2Â [2(4)Â + (15Â â€“ 1) (-5)]
= 15/2Â [8 + 14(-5)]
= 15/2 (8 â€“ 70)
= 15/2 (-62)
= 15(-31)
= -465
11. If the sum of the firstÂ nÂ terms of an AP is 4nÂ âˆ’Â n^{2}, what is the first term (that isÂ S_{1})? What is the sum of first two terms? What is the second term? Similarly find the 3^{rd}, the10^{th}Â and theÂ n^{th}Â terms.
Solution: Given that,
S_{n}Â = 4nÂ âˆ’Â n^{2}
First term,Â aÂ =Â S_{1}Â = 4(1) âˆ’ (1)^{2}Â = 4 âˆ’ 1 = 3
Sum of first two terms =Â S_{2}= 4(2) âˆ’ (2)^{2}Â = 8 âˆ’ 4 = 4
Second term,Â a_{2}Â =Â S_{2}Â âˆ’Â S_{1}Â = 4 âˆ’ 3 = 1
Common difference, dÂ =Â a_{2}Â âˆ’Â aÂ = 1 âˆ’ 3 = âˆ’2
Nth term, a_{n}Â =Â aÂ + (nÂ âˆ’ 1)dÂ
= 3 + (nÂ âˆ’ 1) (âˆ’2)
= 3 âˆ’ 2nÂ + 2
= 5 âˆ’ 2n
Therefore,Â a_{3}Â = 5 âˆ’ 2(3) = 5 âˆ’ 6 = âˆ’1
a_{10}Â = 5 âˆ’ 2(10) = 5 âˆ’ 20 = âˆ’15
Hence, the sum of first two terms is 4. The second term is 1.
The 3^{rd}, the 10^{th}, andÂ the n^{th}Â terms are âˆ’1, âˆ’15, and 5 âˆ’ 2nÂ respectively.
12. Find the sum of first 40 positive integers divisible by 6.
Solution: The positive integers that are divisible by 6 are 6, 12, 18, 24 â€¦.
We can see here, that this series forms an A.P. whose first term is 6 and common difference is 6.
aÂ = 6
dÂ = 6
S_{40}Â =Â ?
By the formula of sum of nth term, we know,
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
Therefore, putting n = 40, we get,
S_{40Â }= 40/2Â [2(6)Â + (40 â€“ 1) 6]
= 20[12 + (39) (6)]
= 20(12 + 234)
= 20 Ã— 246
= 4920
13. Find the sum of first 15 multiples of 8.
Solution: The multiples of 8 are 8, 16, 24, 32â€¦
The series is in the form of AP, having first term as 8 and common difference as 8.
Therefore,Â aÂ = 8
dÂ = 8
S_{15}Â = ?
By the formula of sum of nth term, we know,
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
S_{15}Â = 15/2Â [2(8)Â + (15 â€“ 1)8]
=Â 15/2[6 + (14) (8)]
=Â 15/2[16 + 112]
= 15(128)/2
= 15 Ã— 64
= 960
14. Find the sum of the odd numbers between 0 and 50.
Solution: The odd numbers between 0 and 50 are 1, 3, 5, 7, 9 â€¦ 49.
Therefore, we can see that these odd numbers are in the form of A.P.
Hence,
First term, aÂ = 1
Common difference, dÂ = 2
Last term, lÂ = 49
By the formulas of last term, we know,
lÂ =Â aÂ + (nÂ âˆ’ 1)Â d
49 = 1 + (nÂ âˆ’ 1)2
48 = 2(nÂ âˆ’ 1)
nÂ âˆ’ 1 = 24
nÂ = 25 = Number of terms
By the formula of sum of nth term, we know,
S_{n}Â =Â n/2 (aÂ +Â l)
S_{25}Â = 25/2 (1 + 49)
= 25(50)/2
=(25)(25)
= 625
15.Â A contract on construction job specifies a penalty for delay of completion beyond a certain dateas follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs. 300 for the third day, etc., the penalty for each succeeding day being Rs. 50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days.
Solution:
We can see, that the given penalties are in the form of A.P. having first term as 200 and common difference as 50.
Therefore, aÂ = 200 and dÂ = 50
Penalty that has to be paid if contractor has delayed the work by 30 days =Â S_{30 }
By the formula of sum of nth term, we know,
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
Therefore,
S_{30}= 30/2Â [2(200) + (30 â€“ 1) 50]
= 15Â [400 + 1450]
= 15 (1850)
= 27750
Therefore, the contractor has to pay Rs 27750 as penalty.
16. A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each of the prizes.
Solution: Let the cost of 1^{st}Â prize beÂ Rs.P.
Cost of 2^{nd}Â prize =Â Rs.PÂ âˆ’ 20
And cost of 3^{rd}Â prize =Â Rs.PÂ âˆ’ 40
We can see that the cost of these prizes are in the form of A.P., having common difference as âˆ’20 and first term asÂ P.
Thus, aÂ =Â P and dÂ = âˆ’20
Given that,Â S_{7}Â = 700
By the formula of sum of nth term, we know,
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
7/2Â [2aÂ + (7 â€“ 1)d]Â = 700
aÂ + 3(âˆ’20) = 100
aÂ âˆ’ 60 = 100
aÂ = 160
Therefore, the value of each of the prizes was Rs 160, Rs 140, Rs 120, Rs 100, Rs 80, Rs 60, and Rs 40.
17. In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of class I will plant 1 tree, a section of class II will plant 2 trees and so on till class XII. There are three sections of each class. How many trees will be planted by the students?
Solution:
It can be observed that the number of trees planted by the students is in an AP.
1, 2, 3, 4, 5â€¦â€¦â€¦â€¦â€¦â€¦..12
First term,Â aÂ = 1
Common difference,Â dÂ = 2 âˆ’ 1 = 1
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
S_{12}Â = 12/2Â [2(1)Â + (12 â€“ 1)(1)]
= 6 (2 + 11)
= 6 (13)
= 78
Therefore, number of trees planted by 1 section of the classes = 78
Number of trees planted by 3 sections of the classes = 3 Ã— 78 = 234
Therefore, 234 trees will be planted by the students.
18. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm, â€¦â€¦â€¦ as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (TakeÂ Ï€ = 22/7)
Solution: We know,
Perimeter of a semi-circle = Ï€r
Therefore,
_{P}_{1}Â = Ï€(0.5) = Ï€/2 cm
_{P}_{2}Â = Ï€(1) = Ï€ cm
_{P}_{3}Â = Ï€(1.5) = 3Ï€/2 cm
Where, P_{1,}Â P_{2},Â P_{3}Â are the lengths of the semi-circles.
Hence we got a series here, as,
Ï€/2, Ï€, 3Ï€/2, 2Ï€, â€¦.
P_{1}Â = Ï€/2 cm
P_{2}Â = Ï€ cm
Common difference, dÂ =Â P2-Â P1 =Â Ï€ â€“ Ï€/2 = Ï€/2
First term =Â P_{1}=Â aÂ =Â Ï€/2 cm
By the sum of nth term formula, we know,
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
Therefor, Sum of the length of 13 consecutive circles is;
S_{13}Â =Â 13/2Â [2(Ï€/2)Â + (13Â â€“ 1)Ï€/2]
= Â 13/2Â [Ï€Â + 6Ï€]
=13/2Â (7Ï€)
=Â 13/2 Ã— 7 Ã—Â 22/7
= 143 cm
19. 200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on. In how many rows are the 200 logs placed and how many logs are in the top row?
Solution:Â We can see that the numbers of logs in rows are in the form of an A.P. 20, 19, 18â€¦
For the given A.P.,
First term, aÂ = 20 and common difference, dÂ =Â a_{2}Â âˆ’Â a_{1}Â = 19 âˆ’ 20 = âˆ’1
Let a total of 200 logs be placed inÂ nÂ rows.
Thus, S_{n}Â = 200
By the sum of nth term formula,
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
S_{12}Â = 12/2Â [2(20)Â + (nÂ â€“ 1)(-1)]
400 =Â nÂ (40 âˆ’Â nÂ + 1)
400 =Â nÂ (41 âˆ’Â n)
400 = 41nÂ âˆ’Â n^{2}
n^{2}Â âˆ’ 41nÂ + 400 = 0
n^{2}Â âˆ’ 16nÂ âˆ’ 25nÂ + 400 = 0
nÂ (nÂ âˆ’ 16) âˆ’25 (nÂ âˆ’ 16) = 0
(nÂ âˆ’ 16) (nÂ âˆ’ 25) = 0
Either (nÂ âˆ’ 16) = 0 orÂ nÂ âˆ’ 25 = 0
nÂ = 16 orÂ nÂ = 25
By the nth term formula,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)d
a_{16}Â = 20 + (16 âˆ’ 1) (âˆ’1)
a_{16}Â = 20 âˆ’ 15
a_{16}Â = 5
Similarly, the 25^{th} term could be written as;
a_{25}Â = 20 + (25 âˆ’ 1) (âˆ’1)
a_{25}Â = 20 âˆ’ 24
= âˆ’4
It can be seen, the number of logs in 16^{th}Â row is 5 as the numbers cannot be negative.
Therefore, 200 logs can be placed in 16 rows and the number of logs in the 16^{th}Â row is 5.
20. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato and other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.
A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?
[Hint: to pick up the first potato and the second potato, the total distance (in metres) run by a competitor is 2 Ã— 5 + 2 Ã—(5 + 3)]
Solution: The distances of potatoes from the bucket are 5, 8, 11, 14â€¦, which is in the form of AP.
Given, the distance run by the competitor for collecting these potatoes are two times of the distance at which the potatoes have been kept.
Therefore, distances to be run w.r.t distances of potatoes, could be written as;
10, 16, 22, 28, 34,â€¦â€¦â€¦.
Hence, the first term, aÂ = 10 and dÂ = 16 âˆ’ 10 = 6
S_{10}Â =?
By the formula of sum of nth term, we know,
S_{10}Â = 12/2Â [2(20)Â + (nÂ â€“ 1)(-1)]
= 5[20 + 54]
= 5 (74)
= 370
Therefore, the competitor will run a total distance of 370 m.
Class 10 Maths Chapter 5 Exercise 5.4 Page: 115
1. Which term of the AP : 121, 117, 113, . . ., is its first negative term? [Hint : Find n for an < 0]
Solution: Given the AP series is 121, 117, 113, . . .,
Thus, first term, a = 121
Common difference, d = 117-121= -4
By the nth term formula,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)d
Therefore,
a_{n}Â =Â 121Â + (nÂ âˆ’ 1)(-4)
= 121-4n + 4
=125-4n
To find the first negative term of the series, a_{n }< 0
Therefore,
125-4n < 0
125 < 4n
n>125/4
n>31.25
Therefore, the first negative term of the series is 32^{nd} term.
2. The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.
Solution: From the given statements, we can write,
a_{3 + }a_{7} = 6 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.(i)
And
a_{3 Ã— }a_{7 }= 8 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..(ii)
By the nth term formula,
a_{n}Â =Â aÂ + (nÂ âˆ’ 1)d
Third term, a_{3 }= a + (3 -1)d
a_{3 }= a + 2dâ€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦(iii)
And Seventh term, a7 = a + (7 -1)d
a_{7 }= a + 6d â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..(iv)
From equation (iii) and (iv), putting in equation(i), we get,
a + 2d + a + 6d = 6
2a + 8d = 6
a+4d=3
or
a = 3 â€“ 4d â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦(v)
Again putting the eq. (iii) and (iv), in eq. (ii), we get,
(a + 2d) Ã— (a + 6d) = 8
Putting the value of a from equation (v), we get,
(3 â€“ 4d + 2d) Ã— (3 â€“ 4d + 6d) = 8
(3 â€“ 2d) Ã— (3 + 2d) = 8
3^{2 }â€“ 2d^{2} = 8
9 â€“ 4d^{2} = 8
4d^{2} = 1
d = 1/2 or -1/2
Now, by putting both the values of d, we get,
a = 3 â€“ 4d = 3 â€“ 4(1/2) = 3 â€“ 2 = 1, when d = Â½
a = 3 â€“ 4d = 3 â€“ 4(-1/2) = 3+2 = 5, when d = -1/2
We know, the sum of nth term of AP is;
S_{n}Â =Â n/2Â [2aÂ + (nÂ â€“ 1)d]
So, when a = 1 and d=1/2
Then, the sum of first 16 terms are;
S_{16}Â =Â 16/2Â [2Â + (16Â â€“ 1)1/2] = 8(2+15/2) = 76
And when a = 5 and d= -1/2
Then, the sum of first 16 terms are;
S_{16}Â =Â 16/2Â [2.5+ (16Â â€“ 1)(-1/2)] = 8(5/2)=20
3. A ladder has rungs 25 cm apart. (see Fig. 5.7). The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and the bottom rungs are m apart, what is the length of the wood required for the rungs? [Hint : Number of rungs = -250/25 ].
Solution: Given,
Distance between the rungs of the ladder is 25cm.
Distance between the top rung and bottom rung of the ladder is=m = 100 cm = cm
= 250cm
Therefore, total number of rungs = 250/25 + 1 = 11
As we can see from the figure, the ladder has rungs in decreasing order from top to bottom. Thus, we can conclude now, that the rungs are decreasing in an order of AP.
And the length of the wood required for the rungs will be equal to the sum of the terms of AP series formed.
So,
First term, a = 45
Last term, l = 25
Number of terms, n = 11
Now, as we know, sum of nth terms is equal to,
S_{n} = n/2(a+ l)
S_{n} = 11/2(45+25) = 11/2 (70) = 385 cm
Hence, the length of the wood required for the rungs is 385cm.
4. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x. [Hint : Sx â€“ 1 = S49 â€“ Sx ]
Solution: Given,
Row houses are numbers from 1,2,3,4,5â€¦â€¦.49.
Thus we can see the houses numbered in a row are in the form of AP.
So,
First term, a = 1
Common difference, d=1
Let us say the number of xth houses can be represented as;
Sum of nth term of AP = n/2[2a+(n-1)d]
Sum of number of houses beyond x house = S_{x-1}
= (x-1)/2[2.1+(x-1-1)1]
= (x-1)/2 [2+x-2]
= â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦(i)
By the given condition, we can write,
S_{49 }â€“ S_{x} = {49/2[2.1+(49-1)1]} â€“ {x/2[2.1+(x-1)1]}
= 25(49) â€“ x(x + 1)/2 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.(ii)
As per the given condition, eq.(i) and eq(ii) are equal to each other;
Therefore,
As we know, the number of house cannot be an a negative number. Hence, the value of x is 35.
5. A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of 1 4 m and a tread of 1 2 m. (see Fig. 5.8). Calculate the total volume of concrete required to build the terrace. [Hint : Volume of concrete required to build the first step =].
Solution: As we can see from the given figure, the first step is Â½ m wide, 2^{nd} step is 1m wide and 3^{rd} step is 3/2m wide. Thus we can understand that the width of step by Â½ m each time when height is Â¼ m. And also, given length of the steps is 50m all the time. So, the width of steps forms a series AP in such a way that;
Â½ , 1, 3/2, 2, â€¦â€¦..
Volume of steps = Volume of Cuboid
= Length Ã— Breadth Height
NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progression
This chapter comes under unit 3 algebra and this unit has 20 marks allotted in the examination. Students can expect an average of 3 questions from arithmetic progressions. Along with class 10 examinations this topic is very important from the point of competitive exams.
Sub-topics of class 10 chapter 5 Arithmetic Progression
5.1 Introduction
In this chapter, we shall discuss about patterns which we come across in our day-to-day life in which succeeding terms are obtained by adding a fixed number to the preceding terms. We shall also see how to find their nth terms and the sum of n consecutive terms, and use this knowledge in solving some daily life problems.
5.2 Arithmetic Progressions
The topic describes Arithmetic Progressions, its definition and relatable terms along with fine examples. You will also learn about Finite Arithmetic Progressions and Infinite Arithmetic Progressions. The general form of AP is a, a + d, a + 2d, a + 3d,â€¦
5.3 nth Term of an AP
This topic discusses various methods to determine the nth term of an AP. The concepts are explained with different types of problems solving techniques and finding the nth term of an AP. the examples mentioned in the chapter will help you while solving the exercise problems.
5.4 Sum of First n Terms of an AP
The topics discuss different techniques to find the sum of the first n terms of an AP. it also provides suitable examples which show different techniques to find the sum of the first n terms of AP.Â
5.5 Summary
It gives an overview of the entire chapter and the important topics explained in the entire chapter. By going through the summary part you can cover the entire chapter in few points which help in memorizing the essential concepts.
List of Exercise from class 10 Maths Chapter 5 Arithmetic progression
Exercise 5.1â€“ 4 questions 1 MCQ and 3 descriptive type questions
Exercise 5.2â€“ 20 questions, 1 fill in the blanks, 2 MCQâ€™s, 7 Short answer questions and 10 Long answer questions
Exercise 5.3â€“ 20 Questions 3 fill in the blanks, 4 daily life examples, and 13 descriptive type questions
Exercise 5.4 5 Questions- 5 Long answer questions
This NCERT solution for class 10 Maths is a perfect study material that will help you solve different kinds of problems. Solving this NCERT Solutions will help you understand the topic completely and help you lay greater foundation for future studies.
In this chapter students will discuss pattern in succeeding term obtained by adding a fixed number to the preceding terms. Students also see how to find nth terms and the sum of n consecutive terms. Students will learn arithmetic progression effectively when they solve daily life problems.
This chapter has Arithmetic Progression Derivation of the n^{th}Â term and sum of the first n terms of an A.P. and their application in solving daily life problems. This is one of the important chapters from the point of Class 10 examination. Arithmetic progression is very basic and important topic to study ass almost all the competitive exams will questions on arithmetic progression.
Key Features of NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic progressions
- Has answers to different types of questions such as MCQâ€™S and long answer questions.
- Solving this NCERT solution will make you get well versed with important formulas.
- Act as a basis to solve arithmetic progression problems asked in competitive examination.
- Has answers to all the exercise questions provided in NCERT textbook
- Provide you the necessary practice of solving questions
- You can solve different types of questions with varied difficulty.
- Different examples taken from day to day life will help you understand the topic thoroughly.
Keep visiting BYJUâ€™S to get complete assistance for CBSE class 10 board exams. At BYJUâ€™S, students can get several sample papers, question papers, notes, textbooks, videos, animations and effective preparation tips which can help you to score well in the class 10 exams.
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Frequently Asked Questions on Chapter 5 â€“ Arithmetic Progressions
In which of the following situations, does the list of numbers involved make as arithmetic progression and why?The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each additional km?
We can write the given condition as;
Taxi fare for 1 km = 15
Taxi fare for first 2 kms = 15 + 8 = 23
Taxi fare for first 3 kms = 23 + 8 = 31
Taxi fare for first 4 kms = 31 + 8 = 39
And so onâ€¦â€¦Thus, 15, 23, 31, 39 â€¦ forms an A.P. because every next term is 8 more than the preceding term.
Choose the correct choice in the following and justify 30^{th} term of the A.P: 10, 7, 4, â€¦ is 97, 77,âˆ’77,âˆ’87?
A.P. = 10, 7, 4, â€¦
Therefore, we can find,
First term, a = 10
Common difference, d = a_{2} âˆ’ a_{1 }= 7 âˆ’ 10 = âˆ’3
As we know, for an A.P.a_{n} = a + (n âˆ’ 1) d
Putting the values;
a_{30} = 10 + (30 âˆ’ 1) (âˆ’3)
a_{30} = 10 + (29) (âˆ’3)
a_{30} = 10 âˆ’ 87 = âˆ’77
Hence, the correct answer is option C.
Which term of the A.P. 3, 8, 13, 18, â€¦ is 78?
Given the A.P. series as3, 8, 13, 18, â€¦
Thus,
First term, a = 3
Common difference, d = a_{2} âˆ’ a_{1} = 8 âˆ’ 3 = 5
Let the n^{th} term of given A.P. be 78. Now as we know,
a_{n} = a + (n âˆ’ 1) d
Therefore,
78 = 3 + (n âˆ’ 1) 5
75 = (n âˆ’ 1) 5
(n âˆ’ 1) = 15
n = 16
Hence, 16^{th} term of this A.P. is 78.
Find the sum of the following APs 2, 7, 12 ,â€¦., to 10 terms?
Given, 2, 7, 12 ,â€¦, to 10 terms For this A.P.,
first term, a = 2
And common difference, d = a_{2} âˆ’ a_{1} = 7 âˆ’ 2 = 5
n = 10
We know that,the formula for sum of nth term in AP series is,
S_{n} = n/2 [2a + (n â€“ 1) d]
S_{10} = 10/2 [2(2) + (10 â€“ 1) Ã— 5]
5[4 + (9) Ã— (5)]
5 Ã— 49 = 245
Which term of the AP : 121, 117, 113, . . ., is its first negative term? [Hint : Find n for an < 0]
Given the AP series is 121, 117, 113, . . .,
Thus, first term, a = 121
Common difference, d = 117-121= -4
By the nth term formula,
a_{n} = a + (n âˆ’ 1)d
Therefore,
a_{n} = 121 + (n âˆ’ 1)(-4)
= 121-4n + 4
=125-4n
To find the first negative term of the series, a_{n }< 0
Therefore,
125-4n < 0
125 < 4n
n>125/4
n>31.25
Therefore, the first negative term of the series is 32^{nd} term.