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