The concept of finding the sum of n terms of an A.P. is tested in this exercise. Students with uncleared doubts regarding solving problems of this chapter and others can refer the Selina Solutions for Class 10 Maths. Furthermore, the solutions to this exercise are available in the Concise Selina Solutions for Class 10 Maths Chapter 10 Arithmetic Progression Exercise 10(C), PDF which is given in the links below.

## Selina Solutions Concise Maths Class 10 Chapter 10 Arithmetic Progression Exercise 10(C) Download PDF

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### Access Selina Solutions Concise Maths Class 10 Chapter 10 Arithmetic Progression Exercise 10(C)

**1. Find the sum of the first 22 terms of the A.P.: 8, 3, -2, â€¦â€¦â€¦..**

**Solution: **

Given, A.P. 8, 3, -2, â€¦â€¦â€¦..

Here,

a = 8

d = 3 â€“ 8 = -5

And we know that,

S_{n} = n/2[2a + (n – 1)d]

So,

S_{22} = 22/2[2(8) + (22 – 1)(-5)]

= 11[16 + (21 x -5)]

= 11[16 – 105]

= 11[-89]

= – 979

**2. How many terms of the A.P. :**

**24, 21,Â 18, â€¦â€¦â€¦Â must be taken so that their sum is 78?**

**Solution: **

Given, A.P. 24, 21,Â 18, â€¦â€¦â€¦

Here,

a = 24

d = 21 â€“ 24 = -3 and

S_{n} = 78

We know that,

S_{n} = n/2[2a + (n – 1)d]

So,

78 = n/2[2(24) + (n – 1)(-3)]

78 = n/2[48 â€“ 3n + 3]

156 = n[51 â€“ 3n]

3n^{2} â€“ 51n + 156 = 0

n^{2} â€“ 17n + 52 = 0

n^{2} â€“ 13n â€“ 4n + 52 = 0

n(n – 13) â€“ 4(n – 13) = 0

(n – 4) (n â€“ 13) = 0

So, n – 4 = 0 or n â€“ 13 = 0

Thus, n = 4 or 13

Hence, the required number of terms to be taken can be first 4 or first 13.

**3. Find the sum of 28 terms of an A.P. whose n ^{th}Â term is 8n – 5.**

**Solution: **

Given,

n^{th} term of the A.P. = 8n – 5

So,

a = 8(1) â€“ 5 = 8 â€“ 5 = 3

Here, the last term is the 28^{th} term

l = 8(28) – 5 = 224 â€“ 5 = 219

We know that,

S_{n} = n/2(a + l)

Thus,

S_{28} = 28/2(3 + 219) = 14(222) = 3108

**4. (i) Find the sum of all odd natural numbers less than 50**

**Solution: **

Odd natural numbers less than 50 are:

1, 3, 5, 7, â€¦â€¦.. ,49

This forms an A.P. with a = 1, d = 2 and l = 49

And,

49 = 1 + (n – 1)2

49 = 1 + 2n â€“ 2

50 = 2n

n = 25

We know that,

S_{25} = n/2 [a + l]

= 25/2 [1 + 49]

= 25/2 [50] = 25 x 25

Thus, S_{25} = 625

**(ii) Find the sum of first 12 natural numbers each of which is a multiple of 7.**

**Solution: **

The first 12 natural numbers which are multiples of 7 are:

7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77 and 84

This is an A.P where,

a = 7, d = 7, l = 84 and n = 12

So,

S_{12} = 12/2 [7 + 84]

= 6 [91]

= 546

**5. Find the sum of first 51 terms of an A.P. whose 2 ^{nd}Â and 3^{rd}Â terms are 14 and 18 respectively.**

**Solution: **

Given,

Number of terms of an A.P. (n) = 51

t_{2} = 14 and t_{3} = 18

So, the common difference (d) = t_{3} â€“ t_{2} = 18 â€“ 14 = 4

And,

t_{2} = a + d

14 = a + 4

a = 10

Now, we know that

S_{n} = n/2[2a + (n – 1)d]

Thus,

S_{51} = 51/2[2(10) + (51 – 1)(4)]

= 51/2[20 + 50×4]

= 51/2[20 + 200]

= 51/2[220]

= 51 x 110

= 5610

**6. The sum of first 7 terms of an A.P is 49 and that of first 17 terms of it is 289. Find the sum of first n terms.**

**Solution: **

Given,

S_{7 }= 49

S_{17} = 289

We know that,

S_{n} = n/2[2a + (n – 1)d]

So,

S_{7 }= 7/2[2a + (7 – 1)d] = 49

7[2a + 6d] = 98

2a + 6d = 14

a + 3d = 7 â€¦..(1)

And,

S_{17} = 17/2[2a + (17 – 1)d] = 289

17/2[2a + 16d] = 289

17[a + 8d] = 289

a + 8d = 289/17

a + 8d = 17 â€¦â€¦ (2)

Subtracting (1) from (2), we have

5d = 10

d = 2

Using value of d in (1), we get

a + 3(2) = 7

a = 7 â€“ 6 = 1

Therefore,

S_{n} = n/2[2(1) + (n – 1)(2)]

** = **n/2[2 + 2n – 2]

= n/2[2n]

S_{n }= n^{2}

**7. The first term of an A.P is 5, the last term is 45 and the sum of its terms is 1000. Find the number of terms and the common difference of the A.P.**

**Solution: **

Given,

First term of an A.P. = 5 = a

Last term of the A.P = 45 = l

S_{n} = 1000

We know that,

S_{n} = n/2[a + l]

1000 = n/2[5 + 45]

1000 = n/2[50]

20 = n/2

n = 40

Now,

l = a + (n – 1)d

45 = 5 + (40 – 1)d

40 = 39d

d = 40/39

Therefore, the number of terms is 40 and the common difference is 40/39.