Question

Find the sum to $n$ terms of the series
$n+(n-1)x+(n-2)x^2+...+2x^{n-2}+x^{n-1}$

Answer 1

$P=n+(n-1)x+(n-2)x^2+....2x^{n-2}+x^{n-1}$

$=n+(n-1)x+(n-2)x^2+.....\left[\right(n-2\left)\right(n-2\left)\right]x^{n-2}+[n-(n-1\left)\right]x^{n-1}$

$P=n[1+x+x^2+......x^{n-1}]$ $-x[1+2+3+.....(n-2)+(n-1\left)\right]$

$P=n\sum _{n=1}^{n-1}\left(x^n\right)-x\sum _{n=1}^{n-1}\left(n\right)$

$P=n\sum _{n=1}^{(n-1)}{x}^n-x\left[\frac{n(n-1)}{2}\right]$

$\sum _{n=1}^{n-1}{x}^n=$sum of $(n-1)\tan$ in GP where $a=1$ and $r=x$

$\Rightarrow \sum _{n=1}^{n-1}{x}^n=\frac{a(x^{n-1}-1)}{(x-1)}$

$\Rightarrow P=\frac{na(x^{n-1}-1)}{(x-1)}-x\left[\frac{n(n-1)}{2}\right]$.