Question

Find the sum of the series $1+(1+x)+(1+x+x^2)+...$ to n terms, $x\ne 1$.

Answer 1

$S_n=1+(1+x)+(1+x+x^2)+......$n terms

$x{S}_n=x+(x+x^2)+(x+x^2+x^3)+.......$ n terms

Subtracting above series,

$(1-x)S_n=n-(x+x^2+......+x^n)$

$(1-x)S_n=n-\frac{x(x^n-1)}{x-1}$ ........ as there is a G.P. series

$S_n=\frac{n}{1-x}-\frac{x(x^n-1)}{(x-1{)}^2}$