First slide

Arithmetico-Geometric Progression

Question

The sum of the series $1 + 3 x + 6 x^{2} + 10 x^{3} + \dots \infty$ will be

Moderate

A

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

$\frac{1}{1 - x}$
C

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

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

Solution

Let $S = 1 + 3 x + 6 x^{2} + 10 x^{3} + \dots \infty$
$x S = x + 3 x^{2} + 6 x^{3} + \dots \infty$
On subtracting, we get
$\begin{array}{l} S (1 - x) = 1 + 2 x + 3 x^{2} + 4 x^{3} + \dots \infty \\ x (1 - x) S = x + 2 x^{2} + 3 x^{3} + \dots \infty \end{array}$
Again on subtracting, we get
$S [(1 - x) - x (1 - x)] = (1 + x + x^{2} + x^{3} + \dots \infty)$
$\Rightarrow S [(1 - x) (1 - x)] = \frac{1}{1 - x} \Rightarrow S = \frac{1}{(1 - x)^{3}}$

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