Question

If $S_n=nP+\frac{n(n-1)Q}{2}$, where $Sn$ denotes the sum of the first $n$ terms of an $AP$, then the common difference is

• A. $P+Q$
• B. $2P+3Q$
• C. $2Q$
• D. $Q$

Answer 1

Answer: (D)

$Q$

Given $S_n=nP+\frac{n(n-1)Q}{2}$,

Let the first term of the $A.P$ be $a$ and the common difference be $d$,

$\therefore$ sum of $n$ terms is $S_n=\frac{n}{2}(2a+(n-1)d)$,

$⟹na+\frac{n(n-1)d}{2}$,

Comparing this equation with the given equation gives,

$na+\frac{n(n-1)d}{2}=nP+\frac{n(n-1)Q}{2}$,

$\therefore$ The first term of the $A.P$ is $P$ and the common difference is $Q$.