**Q1. Find the sum of the following given AP’s:**

**(i) 3, 7, 11, 15 . . . . . . . up to 12 terms**.

**Sol. **

Here a = 3, d = (7 – 3) = 4 and n = 12

Since, the sum of first 12 terms is given by: **S _{n} = n2 [2a+(n-1)d]**

Therefore, S_{n} =

= 6×(6 + 44) = 300

**Therefore, the sum of first 12 terms is 300. **

**(ii) –17, –30, –43. . . . . . . . up to 16 terms.**

**Sol.**

Here a = -17, d = [-30 – (-17)] = -13 and n = 16

Since, the sum of first n terms is given by: **S _{n} = n2 [2a+(n-1)d]**

Therefore, S_{n} =

= 8×(-34 – 195) = -1832

**Therefore, the sum of first 16 terms is -1832**

**(iii) 0.8, 1.3, 1.8. . . . . . . up to 50 terms.**

**Sol.**

Here a = 0.8, d = (1.3 – 0.8) = 0.5 and n = 50

Since, the sum of first n terms is given by: **S _{n} = n2 [2a+(n-1)d]**

Therefore, S_{n} =

= 25×(1.6 + 24.5) = 652.5

**Therefore, the sum of first 50 terms is 652.5**

** **

**Q.2 The first term of an A.P is 11 and last term of an A.P is 91. If sum of this A.P is 867, find the total number of terms in that A.P and common difference. **

**Sol. **

Given, a = 11, l = 91 (last term) and S_{n} = 867

Since, **S _{n} = n2 [2a+(n-1)d]**

(or) S_{n} =

(or) S_{n} = _{n }= a +(n – 1)d = l)

Therefore, 867 =

867×2 = 102n

Therefore, **n = 17**

Since l = a + (17 – 1)d (since there are total 17 terms)

Therefore, 91 = 11 + 16d

** d = 5 **

**Therefore, total numbers of terms are 17 and common difference is 5**

** **

**Q.3 First term of an A.P is 17 and last term of an A.P is 407. How many terms are there in an A.P also find its sum if common difference is 13.**

**Sol.**

Given, a = 17, d = 13, **l = 407 (last term) = a _{n}**

Since, **a _{n} = a + (n – 1)d**

Therefore, 407 = 17 + (n – 1)×13

13n = 407 + 13 – 17

Therefore, **n = 31**

Since **S _{n} = n2 [2a + (n – 1)d]**

(or) S_{n} =

(or) S_{n} = _{n }= a +(n – 1)d = l)

Therefore, S_{n} =

S_{n} = 6572

**Therefore, total number of terms in the given A.P are 31 and there sum is 6572.**

**Q4. 2 ^{nd} and 4^{th} terms of an A.P are 30 and 44 respectively. Find the sum of first 45 terms.**

**Sol. **

Given, a_{2} = 30 and a_{4} = 44

Since, a_{n} = a + (n – 1)d

Therefore, a_{2} = a + (2-1)d

(or) 30 = a + d . . . . . . . . . . . (1)

And, a_{4} = a + (4 – 1)d

44 = a + 3d . . . . . . . . . (2)

Subtracting equation (1) from equation (2) we get

14 = 2d

Therefore **d = 7**, on putting value of d in equation (2) we get

44 = a + 3 × 7

Therefore **a = 23**

Since, the sum of first n terms is given by: **S _{n} = n2 [2a+(n-1)d]**

Therefore, for **n = 45**

S_{n} =

S_{n} =

Therefore, S_{n} = 7965

**Hence, the sum of first 45 terms is 7965**

** **

**Q.5 If the sum of first n terms of any particular A.P is given by 2n ^{2 }– 6n. Find first five terms.**

**Sol. **

Given, **S _{n} = 2n^{2} – 6n**

Therefore, S_{1} = -4 (2×1^{2 }– 6×1)

S_{2} = -4 (2×2^{2 }– 6×2)

Similarly, S_{3 }= 0, S_{4} = 8, S_{5 }=20

Since **a _{n} = S_{n} – S_{n –1}**

Therefore, a_{1} = S_{1} = -4

a_{2} = S_{2} – S_{1 }= -4 – (-4) = 0

a_{3} = S_{3} – S_{2 }= 0 – (-4) = 4

a_{4} = S_{4} – S_{3 }= 8 – 0 = 8

a_{5} = S_{5} – S_{6 }= 20-8 = 12

**Therefore, first five terms of an A.P are: -4, 0, 4, 8, 12**

** **

**Q.6 A total sum of Rs 7500 is to be used to give 6 cash prizes to students of Saint Marry School for their overall excellent academic performance. If cost of each prize is Rs 200 less than its preceding prize, find the value of each of the prizes.**

**Sol.**

Let the cost of 1^{st} prize be = a

Given, d = -200 (since each prize is Rs 200 less than its preceding prize)

and S_{n} = 7500

Since, **the sum of first n terms is given by: S _{n} = n2 [2a+(n-1)d]**

Therefore, for **n = 6**

7500 =

7500 = 3(2a – 1000)

2500 = 2a – 1000

**a = 1750 **

Therefore, cost of prizes will be:

**1 ^{st} prize = a = 1750Rs**,

**2 ^{nd} prize = (a – d) = 1550Rs**,

Similarly, **3 ^{rd} prize = 1350Rs**,

**4**,

^{th}prize = 1150Rs**5**and

^{th}prize = 950Rs**6**

^{th}prize = 750Rs**Q.7 In a construction contract there is a penalty for delay in completion of a contract beyond a certain date. For the delay of one day it is Rs 500, for 2 ^{nd} day it is Rs 600, and Rs 700 for the 3rd day and so on, If the penalty for each succeeding day is increased by Rs 100 and the job was delayed for 28 days. Find how much money the contractor has to pay as penalty??**

** **

**Sol.**

Given, a = 500 (since, penalty on 1^{st} day is 500 Rs)

d = 100 (since, penalty increases by Rs 100 for each succeeding day.)

And n = 28

Since, the sum of first n terms is given by: **S _{n} = n2 [2a+(n-1)d]**

Therefore, for n = 6

S_{n} =

= 14(3700)

= 51800 Rs

**Therefore, the contractor paid Rs 51800 as penalty.**

**Q.8 Fauzia took part in a sports event; there was a race in which 8 balloons were placed in a straight line. A basket was placed at a distance of 5m from the first balloon and all 8 balloons are 2m apart. Fauzia started from basket, picks up the nearest balloon and runs back to the basket to drop it and she continues in the same way until all balloons are in the basket. Find the total distance Fauzia covered. **

**Sol.**

To pick up 1^{st} balloon and drop it into basket Fauzia covered distance = 2×5m = 10m

Similarly for 2^{nd} balloon, distance covered by Fauzia = 2(5+2) = 14m

Similarly for 3^{rd} balloon, distance covered by Fauzia = 2(5+2+2) = 18m

Since there are 8 balloons therefore, n = 8

Here, a = 10 and d = (18-14) = 4

Since, the sum of first n terms is given by: **S _{n} = n2 [2a+(n-1)d]**

Therefore, for **n = 8**

S_{n} =

= 4(48)

= 192m

**Therefore, total distance covered by Fauzia = 192m**

** **

**Q.9 The product of three numbers in an A.P is -10 and their sum is 6. Find the numbers.**

**Sol.**

Let the numbers be **(a – d), a, (a + d)**

**Now, according to the given conditions:**

(a – d) + a + (a + d) = 6

3a = 6, therefore a = 2

And, (a – d)× a ×(a + d) = -10

(2 – d)(2)(2 + d) = -10

4 – d^{2} = -5

d^{2} = 9

Therefore d = 3 and d = -3

Now, if d = 3

**Then the numbers are: [ -1, 2, 5 ]**

And, if d = -3

**Then the numbers are: [ 5, 2, -1] **

**Extra Questions**

**LAQ**

QUESTIONS-:

*In an A.P., given*pn=4,q=2,An=−14, , find ‘n’ and p.

**Ans.-** Given-:

As we know that,

Also, we know that

Now, putting the value of

We get

Now, putting the value of p = 10 in (i), we have

10 + 2n = 6

Now putting the value of p = -8 in (i), we get

-8 + 2n = 6

Hence, p = -8 and n = 7

*A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows -: Rs 100 for the first day, Rs 150 for the second day, Rs 400 for the third day, etc.; the penalty for each succeeding day being Rs 50 more than for the preceding day. How much does a delay of 30 days cost the contractor?*

**Ans.-** Since the penalty for each succeeding day is Rs 50 more than for the preceding day. Therefore, the amount of penalty for different days form an A.P. with first term p (= 100) and the common difference q (= 50). We have to find how much a delay of 30 days costs the contractor. In other words, we have to find the sum of 30 terms of the A.P.

Thus, a delay of 30 days will cost the contractor of Rs. 24750.

*A sum of Rs. 560 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 out the value of each of the prizes.*

**Ans.-** Total amount of seven prizes = Rs.560

Let us assume the value of first prize be Rs.p

By the given condition, we have

P, p-20, p-40,…

Now,

Therefore, it is an A.P. with common difference (q) = -20

Let K = p + (p-20) + (p-40) + … upto 7 terms

Here, m = p, q =-20 and n = 7

Now,

Hence, the amount of each prize (in rs.) respectively is-:

140, 140-20, 140-40, 140-60, 140-80, 140-100, 140-120

i.e, Rs.140, Rs.120, Rs.100, Rs.80, Rs.60, Rs.40, Rs.20 respectively.

*A sum of Rs. 1000 is to be used to give ten cash prizes to students of a school for their overall academic performance. If each prize is Rs. 20 less than its preceding prize, then find out the value of each of the prizes.*

**Ans.-** Here, the total amount of ten prizes is = Rs.1000

Now, Let us assume the value of first prize be RS. P

So, by the given condition, we have prizes of denomination (in Rs.) p, p – 20, p – 40, …

Now,

Since,

Therefore, it is an A.P. with common difference (q) = -20

Hence,

Here, m = p, q = -20 and n = 10

Now,

Hence, amount of each prize (in Rs.) respectively is 190, (190-20), (190-40), (190-60), (190-80), (190-100), (190-120), (190-140), (190-160) and (190-180)

i.e., Rs.190, Rs.170, Rs.150, Rs.130, Rs.110, Rs.90, Rs.70, Rs.50, Rs.30 and Rs.10.

*A spiral is made-up of successive semicircles, with centres alternately at P and Q, starting with centre at P, of radii 0.5 cm, 0.1 cm, 1.5 cm, 2.0 cm, …. What is the total length of such a spiral made-up of 13 consecutive semicircles ?*(Takeπ=227)

**Ans.-** Since radii of semicircles are 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm …

Now, the length of semicircle

So now, the length of the second

=

Similarly, we can get the length of other semicircle as,

Let us assume ‘A’ be the total length of all semicircles.

i.e., p=

Here,

Since,

*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. Find out in how many rows are 200 logs placed and how many logs are in the top row ?*

**Ans.-** Suppose 200 logs are stacked in ‘s’ rows.

There are 20 logs in the first row and the number of logs in a row is one less than the number of logs in the preceding row. So, number of logs in various rows form an A.P. with first term p (=20) and the common difference q (= -1). As there are 200 logs in all rows.

Now, If s =25, then number of logs in 25^{th} row is equal to 25^{th} terms of an A.P. with first term 20 and the common difference (-1).

^{th} row = p + 24q = 20 – 24 = -4

Clearly, this is not meaningful.

Thus, logs are placed in 16 rows.

Number of logs in top row = Number of logs in 16^{th} row

= 16^{th} term of an A.P. with p = 20 and q = -1

Hence, there are 5 logs in the top row.

*The sum of 4*^{th}*and 8*^{th}*terms of an A.P. is 24 and the sum of 6*^{th}*and 10*^{th}*terms is 44. Calculate A.P.*

**Ans.**Let us assume ‘p’ be the first term and ‘q’ be the common difference of the required A.P.

Or, 6p + 17q = 12 ….(i)

And,

Or, 4p+15q = 11 ….(ii)

Now, multiplying (i) by 2 and (ii) by 3, we get

12p + 34q = 24 …..(iii)

12p + 45q = 33 ……(iv)

Now, Subracting (iii) from (iv), we have 11q = 9 or q =

From (ii), we have

Hence, the required A.P. is -:

Or,

*The 17*^{th}*term of an A.P. is 5 more than twice its 8*^{th}*If the 11*^{th}*term of the A.P. is 43, then calculate its ‘nth’ term.*

**Ans.-** Given,

P + 10q = 43 ….(i)

And,

P + 16q = 5 + 2[p + 7q]

P + 16q = 5 + 2p + 14q

Subtracting (ii) from (i), we get

p+10q – (p – 2q) = 43 – (-5)

p+10q-p+2q = 43+5

From (iii) and (i), we have

P + 40 = 43

We know that,

Hence, the ‘nth’ term of given A.P. is 4n – 1.

*Determine the common difference of an A.P. whose first term is 5 and the sum of it first four term is half the sum of the next four terms.*

**Ans.-** Given, First term = p = 5

And,

=

= 2 (10 + 3q) = 20 + 6q

So,

60 + 18q = 40 + 28q

**The sum of first six terms of an A.P. is 42. The ratio of its 10**^{th}**term to its 30**^{th}**term is 1 : 3. Determine the first and the 13**^{th}**term of the A.P.**

**Ans.-** Let us assume ‘p’ be the first tern and ‘q’ be the common difference.

Here,

2p+5q = 14 …..(i)

Also,

Subtracting (ii) from (i), we get

5q + 2q = 14

7q = 14

q = 2

Now, from (ii), we have -:

So now,

Therefore, the first term of the given A.P. is 2 and its 13^{th} term is 26.

*If the sum of first ‘s’ terms of an A.P. is ‘n’ and the sum of first n terms is ‘s’, then prove that the sum of its first (s+n) terms is –(s+n).*

**Ans.-** Let us assume ‘p’ be the first term and ‘q’ be the common difference

And,

Subtracting (ii) from (i), we get,

(s – 1 – n + 1)q =

(s – n)q =

Also, from (i), we get -:

And now, the sum of first (s + n) terms

=

=

=

=

=

=

*In an A.P., the sum of first terms is given by -:*

^{th} term of the A.P.

**Ans.-** Here,

Now,

Hence, the common difference (q) = 7 – 4 = 3

*A manufacturer of T.V. sets produced 600 sets in the third year and 700 sets in the seventh year. Assuming that the production increase uniformly by a fixed number of every year, calculate-:*

*(i) The production in the 1*^{st}* year.*

*(ii) The production in the 10*^{th}*year.*

*(iii) The total production in first 7years.*

**Ans.-** Let us assume the number of sets produced in 1^{st} year be ‘x’ and ‘y’ be the increase in population every year.

We are given, x + 2y = 600 …..(i)

And, x + 6y = 700 …..(ii)

Now subtracting equation (i) from (ii), we get

4y = 100 or y = 25

Now, Substituting y = 25 in equation (i), we get

x = 550

(1) Production in the first year = x = 550

(2) Production in 10th year = x+9y = 550+(9)(25) = 550 + 225 = 775

(3) Total production in first 7years = x + (x+y) +(x+2y) + …..+ (x+6y)

We know that,

Sum =

Here, x = 550, y = 25, n = 7

=

=

=

= 4375

__SAQ 2 MARK__

__QUESTIONS-:__

*The amount of money in the account every year, when Rs 5000 is deposited at compound interest at 5% per annum.*

**Ans.-** Here, Principal = Rs.5000

Rate of interest = 5% per annum

Therefore, Annual (A) =

Now, the sequence becomes

Here,

Therefore, the given amount of money in the account every year does not form an A.P.

*In which of the following situations, does the list of numbers involved make an arithmetic progression and why ?*

*(i) The amount of air present in a cylinder when a vaccum pump removes 16 of air remaining in the cylinder at a time.*

*(ii) The cost of digging a well after every metre of digging, when it costs Rs.250 for the first metre and rises by Rs. 100 for each subsequent metre.*

**Ans.-** Let us assume,

Similarly,

So, the list of numbers is p,

Now here,

Hence, it is not an A.P.

*Write first four terms of the A.P. whose first term is 20 and common difference is 20.*

**Ans.-** In this case we have, p= 20, q= 20

Hence, the required four terms are 20, 40, 60, 80.

*4.Write the first four terms of the A.P. whose first term is -1 and the common difference is 12.*

**Ans.-** Here we have, p=-1, q=

So,

Hence, the required four terms are

*Write four terms of the A.P. whose first term is -1.25 and the common difference is -0.25.*

**Ans.-** Here we have, p=-1.25, q=-0.25

So,

Hence, the required four terms are -1.25, -1.50, -1.75, -2.0

*Write first four terms of the A.P. when the first term ‘p’ and the common difference ‘q’ are given as follows-:*

*(i) p = -2, q = 0 (ii) p = 4, q = -3*

**Ans.-** (i)Here, First term = p = -2

Second term = p+q = -2+0 = -2

Third term = p+2q = -2+2(0) =-2

Similarly, Fourth term = p+3q = -2+3(0) =-2

Hence, the required four terms are -2, -2, -2, -2.

(ii) We have, First term = p = 4

Second term =p+q =4+(-3)=1

Third term = p+2q = 4+2(-3)= -2

Similarly, Fourth term = p+3q= 4+3(-3)= -5

Hence, the required four terms are 4, 1, -2, -5.

*For the following A.P., write the first term and the common difference of the following 3, 1, -1, -3, …*

**Ans.-** Here A.P. is 3, 1, -1, -3, …

So, the first term

Now,

i.e.,

So therefore, for the given A.P. common difference (q) = -2

*For the following A.P., write the first term (p*1 ) and the common difference (q)13,53,93,133,…

**Ans.-** Here the given A.P. is

So, First term (

Now,

i.e. ,

So therefore, for the given A.P. common difference

*For the following A.P.s, write the first term (p) and the common difference (q)-:*

*(i) -5, -1, 3, 7, … (ii) 0.6, 1.7, 2.8, 3.9, ….*

**Ans.-** Given here, First term (p) = -5

Now the common difference (q) = -1-(-5) = 3-(-1) = 7-3 = 4

(ii) Given here, First term (p) = 0.6

Similarly, the common difference (q) = 1.7 – 0.6 = 2.8 – 1.7 = 3.9 – 2.8 = 1.1

*Find whether the following list of numbers form an A.P If they form an A.P., find the common difference (q) and write three more terms*p,p2,p3,p4,…

**Ans.-** Here the given list of numbers is

So, first term (

Now,

As,

Hence, therefore the given list of numbers does not form an A.P.

*Find whether the following list of numbers form an A.P. If they form an A.P., then find the common difference (q) and write three more terms 3, 3+*2–√ , 3+22–√ , 3+32–√ ,…

**Ans.-** Here the given list of numbers is -: 3, 3+

Now, first term (

So,

Similarly,

=

So here it is concluded that,

Hence, the given list of numbers form an A.P. in which, q =

Now the next three more terms are -:

And ,

*Find whether the following list of numbers form an A.P. If they form an A.P., find the common difference (q) and write three more terms*2–√,8–√,18−−√,32−−√,…

**Ans.-** Here the given list of numbers is -:

Now here, the first term

So,