If $S_n$ denotes the sum of $n$ terms of an Arithmetic Progression (AP), then its common difference is given by:

Question

If $S_n$ denotes the sum of $n$ terms of an Arithmetic Progression (AP), then its common difference is given by:

$a$. $S_{n} - S_{n-1} - S_{n-2}$

$b$. $S_{n} -3S_{n-1} - 2S_{n-2}$

$c$. $S_{n} + 2S_{n-1} - S_{n-2}$

$d$. $S_{n} - 2S_{n-1} + S_{n-2}$

I know that the sum of $n$ terms of AP having first term $a$ and common difference $d$ is given by: $$S_{n}=\dfrac {n}{2} [2a+(n-1)d]$$

But I couldn't get any idea to solve.

Answer 1

$T(n)=S(n)-S(n-1)$

$T(n-1)=S(n-1)-S(n-2)$

$d=T(n)-T(n-1)$

Answer 2

$$S_n-S_{n-1}-(S_{n-1}-S_{n-2})$$

Answer 3

$$ \begin{align} &S_n& &=\frac d2 n^2+\left(a-\frac d2\right)n\\ &\Delta S_n &=S_n-S_{n-1}&=\frac d2 (2n-1)+\left(a-\frac d2\right)\\ &\Delta^2 S_n&=\color{red}{S_n-2S_{n-1}+S_{n-2}}&\color{red}{=d} \end{align}$$