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

Answer 1

Answer: (E)

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