Question

If $S_n$ denotes the sum of first $n$ terms of an arithmetic progression and $a_n$ denotes the $n^{th}$ terms of the same $A.P$ given $S_n=n^{2}p$; where $p,n,\in N$, then

• A. $a_1=p$
• B. Common different $=2p$
• C. $S_p=p^3$
• D. $a_p=2p^2-p$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $a_1=p$
B Common different $=2p$
C $S_p=p^3$
D $a_p=2p^2-p$

for an arithematic progression where $a_1,a_n,d,S_n$ are the first term, $n^{th}$,common difference,sum of n terms ,then $a_n=a_1+(n-1)d,S_n=\frac{n}{2}[2a_1+(n-1\left)d\right]$.

$S_n=\frac{n}{2}[a_1+a_n]=n^{2}p$

$a_1+a_1+(n-1)d=(2a_1-d)+nd=2np$

by comparing,

$2a_1-d=0,nd=2np$

$d=2p,a_1=p$

$a_p=p+(p-1)\left(2p\right)=2p^2-p$

$S_p=\frac{p}{2}[2p+(p-1\left)\right(2p\left)\right]=p^3$