Question

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

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

Answer 1

The correct option is C

$Q$

Explanation for the correct option

Step 1: Aplly the formual

Sum of $n$ terms of the A.P is,

$S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

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

Step 2: According to the question,

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

Then,

$nP+\frac{n\left(n-1\right)Q}{2}=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

$nP+\frac{n\left(n-1\right)Q}{2}=an+\frac{n\left(n-1\right)d}{2}$

Step 3: On comparing this equation, we get,

$nP=na$

And,

$\frac{n\left(n-1\right)Q}{2}=\frac{n\left(n-1\right)d}{2}$

Therefore, the common difference between the A.P is $Q$.

Hence, the correct option is ‘C’.