Question

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

Options

• S_n − 2S_n−1 + S_n−2

• S_n − 2S_n−1 − S_n−2

• S_n − S_n−2

•  S_n − S_n−1

Answer 1

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

Now, the n^th term of the A.P will be given by the following formula,

`a_n = S_n - S_(n-1) `

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

`a_(n - 1) = S_(n - 1) - S_((n - 1) -1)`

         `=S_(n-1) - S_(n-2)`

Now, the common difference of an A.P. (d) =  ` a_n - a_(n-1)` 

=`(S_n - S_(n-1)) - (S_(n-1) - S_(n - 2))`

=`S_n - S_(n-1) - S_(n-1) +S_(n-2)`

`=S_n - 2 S_(n-1) + S_(n - 2)`

Therefore, `D = S_n - 2S_(n-1) + S_(n -2) `