Question

The sum of first $n$ terms of the series
$1+(1+x)y+(1+x+x^2)y^2+(1+x+x^2+x^3)y^3+...$ is

• A. $\left(\frac{1}{1-x}\right)[\frac{1-y^n}{1-y}-y\left(\frac{1-x^n{y}^n}{1-xy}\right)]$
• B. $\left(\frac{1}{1-x}\right)[\frac{1-y^n}{1-y^2}-x\left(\frac{1-x^n{y}^n}{1-xy}\right)]$
• C. $\left(\frac{1}{1-x}\right)[\frac{1-y^n}{1-y}-x^2\left(\frac{1-x^n{y}^n}{1-xy}\right)]$
• D. $\left(\frac{1}{1-x}\right)[\frac{1-y^n}{1-y}-2x\left(\frac{1-x^n{y}^n}{1-xy}\right)]$
• E. $\left(\frac{1}{1-x}\right)[\frac{1-y^n}{1-y}-x\left(\frac{1-x^n{y}^n}{1-xy}\right)]$

Answer 1

The correct option is E $\left(\frac{1}{1-x}\right)[\frac{1-y^n}{1-y}-x\left(\frac{1-x^n{y}^n}{1-xy}\right)]$

Let $E=1+(1+x)y+(1+x+x^2)y^2+(1+x+x^2+x^3)y^3+...$
$=\frac{1}{(1-x)}\left[\right(1-x)+(1-x^2)y+(1-x^3)y^2+(1-x^4)y^3+...n^{\text{th}}\text{term]}$
[ Multiplying numerator and denominator by $(1-x)$ ]
$=\frac{1}{(1-x)}\left[\right(1+y+y^2+...n^{\text{th}}\text{term]}-x(1+xy+(xy{)}^2+...n^{\text{th}}\text{term]}$
$=\frac{1}{(1-x)}[\frac{1-y^n}{1-y}-x\left(\frac{1-x^n{y}^n}{1-xy}\right)]$