Question

Let $S_{1}, S_{2}, S_{3}$ be the respective sums of first $n, 2 n$ and $3 n$ terms of the same arithmetic progression with $a$ as the first term and $d$ as the common difference. If $R=S_{3}-S_{2}-S_{1}$, then $R$ depends on

• A. $a$ and $d$
• B. $d$ and $n$
• C. $a$ and $n$
• D. $a, d$ and $n$

Answer 1

Answer: (B)

$R =S_{3}-S_{2}-S_{1}$
$=\frac{3 n}{2}(2 a+(3 n-1) d)-\frac{2 n}{2}(2 a+(2 n-1) d)$
$-\frac{n}{2}(2 a+(n-1) d)$
$=2 a\left(\frac{3 n}{2}-\frac{2 n}{2}-\frac{n}{2}\right)+\frac{n d}{2}(3(3 n-1)$
$-2(2 n-1)-1(n-1))$
$=\frac{n d}{2}(4 n)=2 n^{2} d$