Question

Let s denot the infinite sum 2+5x+9x2+14x3+20x4+...... Where |x|<1 then s equals

Answer 1

Answer:

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Answer 2

Answer:

$\text { S equals } \frac{(2-x)}{(1-x)^{3}}$

Given Data:

$2+5 x+9 x^{2}+14 x^{3}+20 x^{4}+\ldots \ldots$

Step 1:

Let us assume $S=2+5 x+9 x^{2}+\ldots$

$S^{\star} x=2 x+5 x^{2}+9 x^{3}+\ldots$

Step 2:

$\begin{array}{l}{S^{\star} x=2 x+5 x^{2}+9 x^{3}+\ldots} \\ {S(1-x)=2+3 x+4 x^{2}+\ldots} \\ {S(1-x)^{\star} x=2 x+3 x^{2}+4 x^{3}+\ldots .}\end{array}$

Step 3:

By Taking x as common factor.

$S(1-x)(1-x)=2+x+x^{2}+x^{3}+\cdots=\frac{(2+x)}{(1-x)}$

Step 4:

So, S = [2(1-x) + x)]  

$\bold{S=\frac{(2-x)}{(1-x)^{3}}}$