Question

The sum of $n$ terms of the following series $1+(1+x)+(1+x+x^2)+$... will be?

• A. $\frac{1-x^n}{1-x}$
• B. $\frac{x(1-x^n)}{1-x}$
• C. $\frac{n(1-x)-x(1-x^n)}{(1-x{)}^2}$
• D. None of these

Answer 1

The correct option is C $\frac{n(1-x)-x(1-x^n)}{(1-x{)}^2}$

Let $S_n=1+(1+x)+(1+x+x^2)+$........

Hence,

$(1-x)S_n=(1-x)+(1-x)(1+x)+(1-x)(1+x+x^2)+$........

or

$(1-x)S_n=(1-x)+(1-x^2)+(1-x^3)(1-x^4)+$........

$=n-(x+x^2+x^3+........)$

$=n-\frac{x(1-x^n)}{1-x}$

Hence,

$S_n=\frac{n}{1-x}-\frac{x(1-x^n)}{(1-x{)}^2}$

$\Rightarrow S_n=\frac{n(1-x)-x(1-x^n)}{(1-x{)}^2}$