Question

the sum of n terms of an AP is 136 and common difference is 4 If the last term is 31 the number of terms

Answer 1

Given,

Sₙ = 136, d = 4, aₙ = 31 ⇒ a + (n-1)d = 31 ⇒ a +4n -4 = 31 ⇒ a + 4n = 35
⇒ a = 35-4n .........(i)

We know,

Sₙ = n/2(a + aₙ)

Or 136 = n/2 (35-4n + 31)

Or 272 = n (66 -4n)

Or 272 = 66n - 4n²

Or 4n² -66n +272 = 0

Or 2n² -33n + 136 = 0

Or 2n² - 17n - 16n + 136 = 0

Or n(2n - 17) + 8(2n - 17) = 0

Or (2n - 17) (n - 8) = 0

Hence,

2n - 17 = 0 or n - 8 = 0

n = 17/2 or n = 8
(Not applicable)

Hence the number of terms is 8.