If sn denotes the sum of n terms of AP whose common difference is d and first term is a, - Maths - Arithmetic Progressions

Question

If sn denotes the sum of n terms of AP whose common
difference is d and first term is a, find
sn-2sn-1+sn-2

Lalit Mehra · Accepted Answer

Given, a and d are the first term and common difference of the A.P.

Sum of n term of the A.P, $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$

$S_{n} - 2 S_{n - 1} + S_{n - 2} = \frac{n}{2} [2 a + (n - 1) d] - 2 \frac{(n - 1)}{2} [2 a + (n - 1 - 1) d] + \frac{(n - 2)}{2} [2 a + (n - 2 - 1) d] = \frac{n}{2} [2 a + (n - 1) d] - \frac{2 (n - 1)}{2} [2 a + (n - 2) d] + \frac{(n - 2)}{2} [2 a + (n - 3) d] = \frac{1}{2} [2 a n + n (n - 1) d - 4 a (n - 1) - 2 (n - 1) (n - 2) d + 2 a (n - 2) + (n - 2) (n - 3) d] = \frac{1}{2} [2 a (n - 2 n + 2 + n - 2) + d (n^{2} - n - 2 n^{2} + 6 n - 4 + n^{2} - 5 n + 6)] = \frac{1}{2} (2 a \times 0 + 2 d) = \frac{1}{2} \times 2 d = d$

Thus, the value of $S_{n} - 2 S_{n - 1} + S_{n - 2}$ is d.

If sn denotes the sum of n terms of AP whose common difference is d and first term is a, find sn-2sn-1+sn-2

If s_ndenotes the sum of n terms of AP whose common difference is d and first term is a, find s_n-2s_n-1+s_n-2