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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
(11x)[1yn1yy(1xnyn1xy)]
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B
(11x)[1yn1y2x(1xnyn1xy)]
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C
(11x)[1yn1yx2(1xnyn1xy)]
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D
(11x)[1yn1y2x(1xnyn1xy)]
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E
(11x)[1yn1yx(1xnyn1xy)]
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Solution

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

Mathematics

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