Question

If $S_n$ denotes the sum of $n$ terms of A.P., then $S_{n+3}-3S_{n+2}+3S_{n+1}-S_n=$.

• A. $2S_n$
• B. $S_{n+1}$
• C. $3S_n$
• D. $0$

Answer 1

The correct option is D $0$

$S_{n+3}-3S_{n+2}+3S_{n+1}-S_n$

$=(S_{n+3}-S_{n+2})-2(S_{n+2}-S_{n+1})+(S_{n+1}-S_n)$

$=\left(T_{n+3}\right)-2\left(T_{n+2}\right)+\left(T_{n+1}\right)$ [In $AP$ series, $S_{n+3}-S_{n+2}=T_{n+3}]$

$=T_{n+3}-2\left\{\right(T_{n+1}+T_{n+3})÷2\}+T_{n+1}$

$=T_{n+3}-T_{n+1}-T_{n+3}+T_{n+1}$

$=0$