Question

If $x = 1 + a + a^2 + .................... $to infinity and $y = 1 + b + b^2 + ................... $to infinity, where a, b are proper fractions, then $1 + ab + a^2b^2 + .....$ to infinity is equal :

• A. $\frac{xy}{x+y-1}$
• B. $\frac{xy}{x-y-1}$
• C. $\frac{xy}{x-y+1}$
• D. $\frac{xy}{x+y+1}$

Answer 1

Answer: (A)

If $a, ar, ar^2, ar^3 .........$ are in G.P., then
sum of infinite $GP=a+ar+.....+\infty=\frac{a}{1-r}$
where 'a' is the first term and 'r' is the common ratio of G.P.
Given $x = 1+ a + a^2 + .....\infty$
This is a GP, with common ratio 'a'.
$\Rightarrow x=\frac{1}{1-a} \Rightarrow x-ax=1 \Rightarrow a=\frac{x-1}{x}$
Again, $y = 1+ b + b^{2} + ......\infty$ This is also a G.P., with common ratio 'b'.
$\Rightarrow y=\frac{1}{1-b} \Rightarrow b=\frac{y-1}{y}$
Now, consider $1+ab+a^{2}b^{2}+.....\infty$
which is again a GP with common ratio 'ab'.
$\therefore $ Sum $-\frac{1}{1-ab}=\frac{1}{1-\frac{x-1}{x}. \frac{y-1}{y}}$
$=\frac{xy}{xy-xy+x+y-1}=\frac{xy}{x+y-1}$