Question

If in an A.P., S_n = n²p and S_m = m²p, where S_r denotes the sum of r terms of the A.P., then S_p is equal to

A. \(\frac{1}{2} p^3\)

B. mn p 

C. p ³ 

D. (m + n) p²

Answer 1

Correct answer is C. P³

Let first term = a and Common difference = d 

∴ According to the question, S_n = n² p 

S_n = n/2 (2a + (n–1) d) = n²p 

2a + (n–1) d = 2np……………….(1) 

And S_m = m²p 

S_m= m/2 (2a + (m–1) d) = m²p 

2mp = (2a + (m–1) d)……………….(2) 

Subtracting 2 from 1 

2a + (n–1) d – 2a – (m–1) d = 2np – 2mp 

d (n–1 –m + 1) = 2p (n– m) d = 2p 

putting value of d in (1) 

2a + (n–1) 2p = 2np 

a + (n–1)p = np 

a = p 

now S_p = p/2 (2a + (p–1) d) 

putting value of a = p and d = 2p 

S_p = p/2 (2p + (p–1) 2p) 

S_p = p³