If the sum of an infinite GP is 4 / 3 and the sum of the series obtained on squaring each term is 16 / 27, then its common ratio is

Question

If the sum of an infinite GP is 4 / 3 and the sum of the series obtained on squaring each term is 16 / 27, then its common ratio is (A) 1 / 2 (B) 1 / 4 (C) 1 / 3 (D) 1 / 5

Tardigrade Admin · Accepted Answer

Let first term is a and common ratio is r then (a/1-r)=(4/3)......(1) And (a2/1-r2)=(16/27)...........(2) by (1)2(2) ⇒ (1+r/1-r)=3 ⇒ r=1 / 2

Q. If the sum of an infinite GP is $4/3$ and the sum of the series obtained on squaring each term is $16/27$ , then its common ratio is

$1/2$

$1/4$

$1/3$

$1/5$

Let first term is a and common ratio is $r$ then
$\frac{a}{1 - r} = \frac{4}{3}$ ......(1)
And $\frac{a ^{2}}{1 - r ^{2}} = \frac{16}{27}$ ...........(2)
by $(1)^{2} (2)$
$\Rightarrow \frac{1 + r}{1 - r} = 3$
$\Rightarrow r = 1/2$

Q. If the sum of an infinite GP is 4/3 and the sum of the series obtained on squaring each term is 16/27, then its common ratio is

1/2

1/4

1/3

1/5