Question

If the last term of an A.P. of 30 terms is 119 and the 8^th term from the end (towards the first term) is 91, then find the common difference of the A.P. Hence, find the sum of all the terms of the A.P.

Answer 1

Given, last term, l = 119

No. of terms in A.P. = 30

8^th term from the end = 91

Let d be a common difference and assume that the first term of A.P. is 119 (from the end)

Since the n^th term of AP is

a_n = l + (n – 1)d

∴ a₈ = 119 + (8 – 1)d

⇒ 91 = 119 + 7d

⇒ 7d = 91 – 119

⇒ 7d = –28

⇒ d = –4

Now, this common difference is from the end of A.P.

So, the common difference from the beginning = –d

= –(–4) = 4

Thus, a common difference of the A.P. is 4.

Now, using the formula

l = a + (n – 1)d

⇒ 119 = a + (30 – 1)4

⇒ 119 = a + 29 × 4

⇒ 119 = a + 116

⇒ a = 119 – 116

⇒ a = 3

Hence, using the formula for the sum of n terms of an A.P.

i.e., S_n = `"n"/2[2"a" + ("n" - 1)"d"]`

S₃₀ = `30/2[2 xx 3 + (30 - 1) xx 4]`

= 15 (6 + 29 × 4)

= 15 (6 + 116)

= 15 × 122

= 1830

Therefore, the sum of 30 terms of an A.P. is 1830.