Q. If $x = 1 + a + a^{2} + ...\infty$ and $y = 1 + b + b^{2} + ...\infty$ where $a$ and $b$ are proper fractions, then $1 + ab + a^{2}b^{2} + ...\infty$ equals
A) $\frac{xy}{y + x - 1}$
B) $\frac{x + y}{x - y}$
C) $\frac{x^{2} + y^{2}}{x - y}$
D) None of these
Explanation
(a)
Here, $x = \frac{1}{1 - a},\ y = \frac{1}{1 - b}$
$\Rightarrow \ a = \frac{x - 1}{x},\ \ \ b = \frac{y - 1}{y}$
$\therefore 1 + ab + a^{2}b^{2} + ... = \frac{1}{1 - ab} = \frac{xy}{x + y - 1}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.