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8. Sequences and Series
medium
$x = 1 + a + {a^2} + ....\infty \,(a < 1)$ $y = 1 + b + {b^2}.......\infty \,(b < 1)$ Then the value of $1 + ab + {a^2}{b^2} + ..........\infty $ is
A
$\frac{{xy}}{{x + y - 1}}$
B
$\frac{{xy}}{{x + y + 1}}$
C
$\frac{{xy}}{{x - y - 1}}$
D
$\frac{{xy}}{{x - y + 1}}$
Solution
(a) Since the series are $G.P.$, therefore
$x = \frac{1}{{1 – a}}$
$\Rightarrow a = \frac{{x – 1}}{x}$ and $y = \frac{1}{{1 – b}}$
$\Rightarrow b = \frac{{y – 1}}{y}$
$\therefore $$1 + ab + {a^2}{b^2} + ……….\infty = \frac{1}{{1 – ab}}$
$ = \frac{1}{{1 – \frac{{x – 1}}{x}.\frac{{y – 1}}{y}}} = \frac{{xy}}{{x + y – 1}}$.
Standard 11
Mathematics
