Question

Problems Set at an Oral Examination.
Simplify the following expressions.
$x+2x^2+3x^3+...+n{x}^n.$

Answer 1

$S=x+2x^2+3x^3+...+n{x}^n$

$xS=$ $x^2+2x^2+3x^4+...+(n-1)x^n+n{x}^{n+1}$

$⟹S-xS=(x+x^2+x^3+...x^n)-n{x}^{n+1}$

$⟹(1-x)S=\frac{x(x^n-1)}{x-1}-n{x}^{n+1}$

$⟹S=\frac{x(1-x^n)}{(1-x{)}^2}-\frac{n{x}^{n+1}}{1-x}$