Question

Sum to $infinite$ terms the following series:
$1+4x+7x^2+10x^3+...,|x|<1$.

Answer 1

$S=1+4x+7x^2+10x^3+...$ $..................\left(1\right)$

$Sx=x+4x^2+7x^3+....$ $...............\left(2\right)$

$e{q}^n\left(1\right)–e{q}^n\left(2\right)$, we get

$(1–x)S=1+3x+3x^2+3x^3+\dots ...$ $.........\left(3\right)$

This is a $G.P$. of common ratio $=x$

So, using sum of $G.P.=\frac{a}{(1-r)}$

Here, $a=firstterm$ and $r=commonratio$

So,

$(1–x)S=1+\frac{3x}{(1–x)}$

$(1–x)S=\frac{1-x+3x}{(1–x)}$

$S=\frac{1+2x}{(1–x{)}^2}$

Hence, this is the answer.