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8. Sequences and Series
easy
If the sum of an infinite $G.P.$ and the sum of square of its terms is $3$, then the common ratio of the first series is
A
$1$
B
$\frac{1}{2}$
C
$\frac{2}{3}$
D
$\frac{3}{2}$
Solution
(b) Let the first series be $a + ar + a{r^2} + ………$
then the second series is ${a^2} + {a^2}{r^2} + {a^2}{r^4} + ……….$
their sums are given as $3$. So, we have
$\frac{a}{{1 – r}} = 3$ or $a = 3(1 – r)$
and$\frac{a^2}{{1 – r^2}} = 3$ or ${a^2} = 3(1 – {r^2})$
Eliminating $a,\;{\left\{ {3\,(1 – r)} \right\}^2} = 3\,(1 – {r^2})$
$ \Rightarrow $$3\,(1 – r) = (1 + r)$,
$ \Rightarrow $$4r = 2$ or $r = \frac{1}{2}$.
Standard 11
Mathematics
