First slide

Geometric progression

Question

If $x = 1 + a + a^{2} + \dots (| a | < 1)$ and $y = 1 + b + b^{2}$ $+ \dots (| b | < 1)$ then some of the series $1 + a b + a^{2} b^{2} + \dots$ is

Moderate

A

$\frac{x y}{x - y - 1}$
B

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

$\frac{x y}{x + y + 1}$
D

$\frac{x y}{x + y - 1}$

Solution

$\begin{array}{l} x = 1 + a + a^{2} + \dots = \frac{1}{1 - a} \\ y = \frac{1}{1 - b} \\ \Rightarrow a = 1 - 1 / x, b = 1 - 1 / y \end{array}$
Now $1 + a b + a^{2} b^{2} + \dots$
$\begin{array}{l} = \frac{1}{1 - a b} = \frac{1}{1 - (1 - 1 / x) (1 - 1 / y)} \\ = \frac{x y}{x + y - 1} \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App