If Sn denotes the sum to n terms of an A P and p2Sp = pSp2 for each p∈ - Maths - Sequences and Series

Question

If Sn denotes the sum to n terms of an A P and
p2Sp = pSp2 for each
p∈ N then Sn is
(A) depends on d only (B) divisible by n2 (C)
divisible by n (D) divisible by (n-1) Give the process

Ajanta Trivedi · Accepted Answer

let the first term of AP be a and the common difference be d
sum of n terms = n2 2a+(n-1)d
if the given equation is:
p2Sp=pSp2pSp
p-Sp=0⇒p≠0 and Sp≠0Sp=p
thus Sn must be divisible by n

hope this helps you

If Sn denotes the sum to n terms of an A.P. and p2Sp = pSp2 for each p∈ N then Sn is (A) depends on d only (B) divisible by n2 (C) divisible by n (D) divisible by (n-1) .Give the process.

If S_n denotes the sum to n terms of an A.P. and p²S_p = pS_p² for each p∈ N then S_n is

(A) depends on d only (B) divisible by n² (C) divisible by n (D) divisible by (n-1) .Give the process.