Question

1 . The sum of the series 1 + 3x + 6$x^2$ + 10 $x^3$ + ....... ∞ will be

• A. $\frac{1}{(1-x{)}^2}$
• B. $\frac{1}{1-x}$
• C. $\frac{1}{(1+x{)}^2}$
• D. $\frac{1}{(1-x{)}^3}$

Answer 1

The correct option is D

$\frac{1}{(1-x{)}^3}$

Let S = 1 + 3x + 6$x^2$ + 10 $x^3$ + ....... ∞

⇒ x.S = x + 3$x^2$ + 6$x^3$ + ..........∞

Subtracting S(1-x) = 1 + 2x + 3$x^2$ + 4$x^3$ + ........∞

⇒ x(1 - x)S = x + 2$x^2$ + 3$x^3$ + .........∞

Again subtracting,

⇒ S[(1 - x) - x(1 - x)] = 1 + x + $x^2$ + $x^3$ + ........... ∞

⇒ S[(1 - x)(1 - x)] = $\frac{1}{1-x}$ ⇒ S = $\frac{1}{(1-x{)}^3}$