Question

The sum of first p terms of an AP is $(a{p}^2+bp)$. What is the common difference of the AP?
(a) a
(b) 2a
(c) (a + b)
(d) 3a + b

Answer 1

( b) 2a

Given: S_p = (ap² + bp) ...(i)
Replacing p by (p - 1) in (i), we get:
S_p-1 = a(p - 1)² + b(p - 1)
= a(p² - 2p + 1) + bp - b
= ap² - 2ap + a + bp - b
∴​ T_p = ( S_p - S_p-1)
​ = (ap² + bp) - (ap² - 2ap + a + bp - b )
= 2ap - a + b
∴ p^th term, T_p = (2ap - a + b) ...(ii)

Putting p = 1 in (ii), we get:
T₁= (2a ⨯ 1) - a + b = a + b

Putting p = 2 in (ii), we get:
T₂ = (2a ⨯ 2) - a + b = 3a + b
∴ Common difference, d = T₂ - T₁ = 3a + b - a - b = 2a