Question

For any $AP$, $S_n-2S_{n-1}+S_{n-2}=$ _________

Answer 1

To find the value of $S_n-2S_{n-1}+S_{n-2}$

In an $A.P$ we have

$S_n=\frac{n}{2}[a+l]$

where $n$ is the number of terms

$a$ is the first term

$l$ is the last term.

Now $2S_{n-1}=2\frac{(n-1)}{2}[a+l]S_{n-1}=\frac{(n-2)}{2}[a+l]\therefore S_n-2S_{n-1}+S_{n-2}=\frac{n}{2}[a+l]-\frac{2(n-1)}{2}[a+l]+\frac{(n-2)}{2}[a+l]=(a+l)[\frac{n}{2}-\frac{2(n-1)}{2}+\frac{n-2}{2}]=(a+l)\left[\frac{n-2n+2+n-2}{2}\right]=(a+l)\times 0=0$