In an AP, if S n = 3n 2 + 5n and a k = 164, then find the value of k.

Question

Philoid · Accepted Answer

S_n = 3n² + 5n

S_{n - 1} = 3(n - 1)² + 5(n - 1)

= 3(n² - 2n + 1) + 5n - 5

= 3n² - 6n + 3 + 5n - 5

= 3n² - n - 2

S_n - S_{n - 1} = 3n² + 5n - (3n² - n - 2)

(a₁ + a₂ + a₃ + - - - + a_{n - 1} + a_n) - (a₁ + a₂ + a₃ + - - - + a_{n - 1}) = 6n + 2

a_n = 6n + 2

then

a_k = 6k + 2 = 164

6k = 164 - 2 = 162

k = 27