In an A.P, if $ S_{n} = 3n^2 + 5n $ and $ a_k = 164 $

In an A.P, if $ S_{n} = 3n^2  + 5n $ and $ a_k = 164 $, find the value of k

Answer

S_n = 3n² + 5n

n = 1, S₁ (1^st term) = 3(1)² + 5(1) = 8

n = 2, S₂ (sum of 1^st two terms) = 3(2)² + 5(2) = 22

a₁ + a₂ = 22

⇒ a₂ = 22 - 8 = 14

d = a₂ – a₁ = 14 – 8 = 6

a_k = 164

⇒ 8 + (k - 1)6 = 164

k = 27