First slide

Geometric progression

Question

$If x = 1 + a + a^{2} + \dots . \infty, where | a | < 1 and y = 1 + b + b^{2} + \dots \infty where |b| <1, then 1 + ab$ $+ 1 + a b + a^{2} b^{2} + \dots \infty =$

Moderate

A

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

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

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

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

Solution

$x = \frac{1}{1 - a} and y = \frac{1}{1 - b} \Rightarrow a = \frac{x - 1}{x} a n d b = \frac{y - 1}{y} ∴ 1 + a b + a^{2} b^{2} + . . . . . \infty = \frac{1}{1 - a b} = \frac{x y}{x + y - 1}$

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