Question

The sum of the series 1 + 2x + 3 $x^2$ + 4 $x^3$ + ..........upto n

terms is

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

Answer 1

The correct option is A

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

Let $S_n$ be the sum of the given series to n terms, then

$S_n$ = 1 + 2x + 3 $x^2$ + 4 $x^3$ + ....... + n $x^{n-1}$ ..........(i)

x$S_n$ = x + 2 $x^2$ + 3 $x^2$ + ...........+ n $x^n$ ..........(ii)

Subtracting (ii) from (i), we get

(1-x)$S_n$ = 1 + x + $x^2$ + $x^3$ + ......... to n terms -n $x^n$

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

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