# The common difference of an A.P., the sum of whose n terms is Sn, is

Question:

The common difference of an A.P., the sum of whose $n$ terms is $S_{n}$, is

(a) $S_{n}-2 S_{n-1}+S_{n-2}$

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

(c) $S_{n}-S_{n-2}$

(d) $S_{n}-S_{n-1}$

Solution:

Here, we are given an A.P. the sum of whose n terms is Sn. So, to calculate the common difference of the A.P, we find two consecutive terms of the A.P.

Now, the nth term of the A.P will be given by the following formula,

$a_{n}=S_{n}-S_{n-1}$

Next, we find the (− 1)th term using the same formula,

$a_{n-1}=S_{e-1}-S_{(\pi-1)-1}$

$=S_{n-1}-S_{n-2}$

Now, the common difference of an A.P. $(d)=a_{n}-a_{n-1}$

$=\left(S_{n}-S_{n-1}\right)-\left(S_{n-1}-S_{n-2}\right)$

$=S_{n}-S_{n-1}-S_{n-1}+S_{a-2}$

$=S_{n}-2 S_{n-1}+S_{n-2}$

Therefore, $d=S_{n}-2 S_{n-1}+S_{n-2}$

Hence the correct option is (a).