R S Aggarwal Solutions for Class 10 Maths exercise 5d are available here. Students can get detailed step-by-step explanations to all the questions listed under exercise 5d of Arithmetic Progression of the Class 10 R S Aggarwal Textbook. Download pdf of Class 10 Maths Chapter 5 R S Aggarwal Solutions and accelerate your knowledge.

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Exercise 5A Solutions: 30 Questions (Short Answers)

Exercise 5B Solutions: 16 Questions (Short Answers)

Exercise 5C Solutions: 15 Questions (Short Answers)

**Exercise 5D Page No: 289**

**Question 1: The first three terms of an AP are respectively (3y â€“ 1), (3y + 5) and (5y + 1), find the value of y. **

**Solution:**

Given: (3y â€“ 1), (3y + 5) and (5y+ 1) are in AP

So, (3y + 5) â€“ (3y â€“ 1) = (5y + 1) â€“ (3y + 5)

2 (3y + 5) = (5y + 1) + (3y â€“ 1)

6y + 10 = 8y

8y â€“ 6y = 10

2y = 10

Or y = 5

The value of y is 2.

**Question 2: If k, (2k â€“ 1) and (2k + 1) are the three successive terms of an AP, find the value of k. **

**Solution:**

Given: k, (2k â€“ 1) and (2k + 1) are the three successive terms of an AP.

So, (2k â€“ 1) â€“ k = (2k + 1) â€“ (2k â€“ 1)

2 (2k â€“ 1) = 2k + 1 + k

4k â€“ 2 = 3k + 1

4k â€“ 3k = 1 + 2

or k = 3

The value of k is 3.

**Question 3: If 18, a, (b â€“ 3) are in AP, then find the value of (2a â€“ b). **

**Solution:**

Given: 18, a, (b â€“ 3) are in AP

a â€“ 18 = b â€“ 3 â€“ a

a + a â€“ b = -3 + 18

2a â€“ b = 15. Answer!

**Question 4: If the numbers a, 9, b, 25 form an AP, find a and b. **

**Solution:**

Given: a, 9, b, 25 are in AP.

So, 9 â€“ a = b â€“ 9 = 25 â€“ b

b â€“ 9 = 25 â€“ b

b + b = 22 + 9 = 34

or b = 17

And,

a â€“ b = a â€“ 9

9 + 9 = a + b

a + b = 18

a + 17 = 18

or a = 1

Answer: a = 18, b= 17

**Question 5: If the numbers (2n â€“ 1), (3n + 2) and (6n â€“ 1) are in AP, find the value of n and the numbers.**

**Solution:**

Given: (2n â€“ 1), (3n + 2) and (6n â€“ 1) are in AP

So, (3n + 2) â€“ (2n â€“ 1) = (6n â€“ 1) â€“ (3n + 2)

(3n + 2) + (3n + 2) = 6n â€“ 1 + 2n â€“ 1

6n + 4 = 8n â€“ 2

8n â€“ 6n = 4 + 2

Or n = 3

Numbers are:

2 x 3 â€“ 1 = 5

3 x 3 + 2 = 11

6 x 3 â€“ 1 = 17

Answer: (5, 11, 17) are required numbers.

**Question 6: How many three-digit natural numbers are divisible by 7?**

**Solution:**

3-digit natural numbers: 100, 101,â€¦â€¦.. 990 and

3-digit natural numbers divisible by 7: 105, 112, 119, 126, â€¦, 994\

Here, a = 105, d= 7, l = 994

a_n = (l) = a + (n â€“ 1) d

994 = 105 + (n â€“ 1) x 7

994 â€“ 105 = (n â€“ 1) 7

n â€“ 1 = 127

n = 128

Answer: There are 128 required numbers.

**Question 7: How many three-digit natural numbers are divisible by 9?**

**Solution:**

3-digit numbers: 100, 101,â€¦â€¦.,999

3-digit numbers divisible by 9 : 108, 117, 126, 135, â€¦, 999

Here, a = 108, d= 9, l = 999

a_n (l) = a + (n â€“ 1) d

999 = 108 + (n â€“ 1) x 9

(n â€“ 1) x 9 = 999 â€“ 108 = 891

n â€“ 1 = 99

n = 100

**Question 8: If the sum of first m terms of an AP is (2mÂ² + 3m) then what is its second term?**

**Solution:**

Sum of first m terms of an AP = 2mÂ² + 3m (given)

S_m = 2mÂ² + 3m

Sum of one term = S_1 = 2(1)Â² + 3 x 1 = 2 + 3 = 5 = first term

Sum of first two terms = S_2 = 2(2)Â² + 3 x 2 = 8 + 6=14

Sum of first three terms = S_3 = 2(3)Â² + 3 x 3 = 18 + 9 = 27

Now,

Second term = a_2 = S_2 â€“ S_1 = 14 â€“ 5 = 9

**Question 9: What is the sum of first n terms of the AP, a, 3a, 5a,………….**

**Solution:**

AP is a, 3a, 5a,…..

Here, a = a, d = 2a

Sum = S_n = n/2 [2a + (n-1)d]

= n/2[2a + 2an – 2a]

=an^2

**Question 10.: What is the 5th term from the end of the AP 2, 7, 12, â€¦â€¦â€¦ 47?**

**Solution:**

Given AP is 2, 7, 12, 17, â€¦â€¦ 47

Here, a = 2, d = 7 â€“ 2 = 5, l = 47

nth term from the end = l â€“ (n â€“ 1 )d

5th term from the end = 47 â€“ (5 â€“ 1) 5 = 47 â€“ 4 x 5 = 27

**Question 11: If an denotes the nth term of the AP 2, 7, 12, 17, â€¦, find the value of (a_30 â€“ a_20).**

**Solution:**

Given AP is 2, 7, 12, 17,â€¦â€¦..

Here, a = 2, d = 7 â€“ 2 = 5

Now,

a_n = a + (n â€“ 1) d = 2 + (n â€“ 1) 5 = 5n â€“ 3

a_30 = 2 + (30 â€“ 1) 5 = 2 + 145 = 147 and

a_20 = 2 + (20 â€“ 1) 5 = 2 + 95 = 97

a_30 â€“ a_20 = 147 â€“ 97 = 50

**Question 12: The nth term of an AP is (3n + 5). Find its common difference.**

**Solution:**

Nth term = a_n = 3n + 5 (given)

a_(n-1) = 3 (n â€“ 1) + 5 = 3n + 2

Common difference = d = a_n â€“ a_(n-1)

= (3n + 5) â€“ (3n + 2)

= 3n + 5 â€“ 3n â€“ 2

= 3

Therefore, common difference is 3.

**Question 13: The nth term of an AP is (7 â€“ 4n). Find its common difference.**

**Solution:**

Nth term = a_n = 7 â€“ 4n

a_(n-1) = 7 â€“ 4(n â€“ 1) = 11 â€“ 4n

Common difference = d = a_n â€“ a_(n-1)

= (7 â€“ 4n) â€“ (11 â€“ 4n)

= 7 â€“ 4n â€“ 11 + 4n

= -4

Therefore, common difference is -4.

**Question 14: Write the next term of the AP âˆš8, âˆš18, âˆš32, ……..**

**Solution:**

Given AP is âˆš8, âˆš18, âˆš32,……..

Above AP can be written as:

2âˆš2 , 3âˆš2, 4 âˆš2, ……

Here a = 2âˆš2 and d = âˆš2

Next term = 4âˆš2 + âˆš2 = 5âˆš2 = âˆš50

**Question 15: Write the next term of the AP âˆš2, âˆš8, âˆš18,….**

**Solution: **

Given AP is âˆš2, âˆš8, âˆš18,….

Can be written as:

âˆš2, 2âˆš2, 3âˆš2,…..

First term = âˆš2

Common difference = 2âˆš2 – âˆš2 = âˆš2

Next term = 3âˆš2 + âˆš2 = 4âˆš2 = âˆš32

**Question 16: Which term of the AP 21, 18, 15,…. is zero?**

**Solution: **

Given AP is 21, 18, 15,….

First term = a = 21

Common difference = d = 18-21 = -3

Last term = l = 0

l = a + (n â€“ 1) d

0 = 21 + (n â€“ 1)(-3)

0 = 21 â€“ 3n + 3

24 â€“ 3n = 0

Or n = 8

Answer: Zero is the 8th term.

**Question 17: Find the sum of first n natural numbers.**

**Solution:**

First n natural numbers: 1, 2, 3, 4, 5, â€¦, n

Here, a = 1, d = 1

Sum = S_n = n/2 [2a + (n-1)d]

= n/2 [2(1) + (n-1)(1)]

= n(n+1)/2

**Question 18: Find the sum of first n even natural numbers.**

**Solution:**

First n even natural numbers: 2, 4, 6, 8, 10,….,n

Here, a = 2, d = 4 â€“ 2 = 2

Sum = S_n = n/2 [2a + (n-1)d]

= n/2 [2(2) + (n-1)(2)]

= n(n+1)

**Question 19: The first term of an AP is p and its common difference is q. Find its 10th term.**

**Solution:**

Given:

First term =a = p and

Common difference = d =q

Now,

a_10 = a + (n â€“ 1) d

= p + (10 â€“ 1)q

= (p + 9q)

**Question 20: If 4/5, a, 2 are in AP, find the value of a.**

**Solution:**

AP terms: 4/5, a, 2 (given)

Then,

a – 4/5 = 2 – a

a = 7/5

**Question 21: If (2p + 1), 13, (5p â€“ 3) are in AP, find the value of p.**

**Solution:**

Given, 2p + 1, 13, 5p â€“ 3 are in AP

Then,

13 â€“ (2p + 1) = (5p â€“ 3) â€“ 13

13 â€“ 2p â€“ 1 = 5p â€“ 3 â€“ 13

12 â€“ 2p = 5p â€“ 16

p = 4

The value of p is 4.

**Question 22: If (2p â€“ 1), 7, 3p are in AP, find the value of p. **

**Solution:**

Given, (2p â€“ 1), 7, 3p are in AP

Then,

7 â€“ (2p â€“ 1) = 3p â€“ 7

7 â€“ 2p + 1 = 3p â€“ 7

5p = 15

p = 3

The value of p is 3.

**Question 23: If the sum of first p terms of an AP is (apÂ² + bp), find its common difference.**

**Solution:**

Sum of first p terms = S_p = (apÂ² + bp)

Sum of one term = S_1 = a(1)Â² + b(1) = a+b = first term

Sum of first two terms = S_2 = a(2)Â² + b x 2 = 4a + 2b

We know that, second term = a_2 = S_2 – S_1

= (4a + 2b) – (a + b)

= 3a + b

Now, d = a_2 – a_1

= 3a + b – (a+b)

= 2a

Answer: Common difference is 2a.

**Question 24: If the sum of first n terms is (3nÂ² + 5n), find its common difference.**

**Solution:**

Sum of first n terms = S_n = (3nÂ² + 5n)

Sum of one term = S_1 = 3(1)Â² + 5(1) = 8 = first term

Sum of first two terms = S_2 = 3(2)Â² + 5(2) = 22

We know that, second term = a_2 = S_2 – S_1

=22 – 8

= 14

=> a_2 = 14

Now, d = a_2 – a_1

= 14 – 8 = 6

Answer: Common difference is 6.

**Question 25: Find an AP whose 4th term is 9 and the sum of its 6th and 13th terms is 40.**

**Solution:**

Let a be the first term and d be the common difference.

Given:

4th term = a_4 = 9

Sum of 6th and 13th terms = a_6 + a_13 = 40

Now,

a_4 = a + (4-1)d

9 = a + 3d

a = 9 – 3d ….(1)

And

a_6 + a_13 = 40

a + 5d + a + 12d = 40

2a +17d = 40

2(9 – 3d) + 17d = 40 (using (1))

d = 2

From (1): a = 9 – 6 = 3

Required AP = 3,5,7,9,…..

**Question 26: What is the common difference of an AP in which a_27 â€“ a_7 = 84? **

**Solution:**

Given: a_27 â€“ a_7 = 84

[a + 26d] – [a + 6d] = 8420d = 84

d = 4.2

**Question 27: If 1 + 4 + 7 + 10 + â€¦ + x = 287, find the value of x.**

**Solution:**

Given: 1 + 4 + 7 + 10 + â€¦ + x = 287

Here a = 1, d = 3 and S_n = 287

Sum = S_n = n/2 [2a + (n-1)d]

287 = n/2 [2 + (n-1)3]

574 = 3n^2 – n

Which is a quadratic equation.

Solve 3n^2 – n – 574 = 0

3n^2 – 42n + 41n – 574 = 0

3n(n – 14) + 41(n-14) = 0

(3n + 41)(n-14) = 0

Either (3n + 41) = 0 or (n-14) = 0

n = -41/3 or n = 14

Since number of terms cannot be negative, so result is n = 14.

=> Total number of terms in AP are 14.

Which shows, x = a_14

or x = a + 13d

or x = 1 + 39

or x = 40

The value of x is 40.

**R S Aggarwal Solutions for Class 10 Maths Chapter 5 Arithmetic Progression Exercise 5D**

Class 10 Maths Chapter 5 Arithmetic Progression Exercise 5D is based on the following topics:

- General term of an AP: a_n = a + (n-1)d
- nth term from the end of an AP
- Arithmetic mean between two numbers
- Sum of n terms of an AP