Question

The sum of n terms of an AP is given by $S_n=(2n^2+3n)$. What is the common difference of the AP?
(a) 3
(b) 4
(c) 5
(d) 9

Answer 1

( b) 4
Given: S_n = (2n² + 3n) ...(i)
Replacing n by (n - 1) in (i), we get:
S_n-1 = 2(n - 1)² + 3(n - 1)
= 2(n² - 2n + 1) + 3n - 3
= 2n² - n - 1
∴​ T_n = ( S_n - S_n-1)
​ = (2n² + 3n) - (2n² - n - 1) = 4n + 1
∴ n^th term, T_n = (4n + 1) ...(ii)

Putting n = 1 in (ii), we get:
T₁ = (4 ⨯ 1) + 1 = 5

Putting n = 2 in (ii), we get:
T₂ = (4 ⨯ 2) + 1 = 9
∴ Common difference, d = T₂ - T₁ = 9 - 5 = 4