If the sum of an infinite GP $\mathrm{a}, \mathrm{ar}, \mathrm{ar}^{2}, \mathrm{ar}^{3}, \ldots \ldots$ is 15 and the sum of the squares of its each term is 150, then the sum of $a r^{2}, a r^{4}, a r^{6}, \ldots$. is:

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Sequences and Series

If the sum of an infinite GP

a, ar, ar^{2}, ar^{3}, \dots\dots

is 15 and the sum of the squares of its each term is 150, then the sum of

a r^{2}, a r^{4}, a r^{6}, \dots

. is:

Answer: (A)

For

∣ r ∣ < 1

the two given conditions are

\frac{a}{1 - r} = 15, \frac{a ^{2}}{1 - r ^{2}} = 150.

Writing

1 - r^{2} = (1 - r) (1 + r)

and using the first relation in the second,

\frac{a}{1 - r} \cdot \frac{a}{1 + r} = 150 \Rightarrow \frac{15 a}{1 + r} = 150 \Rightarrow a = 10 (1 + r) .

Together with

a = 15 (1 - r)

this gives

15 - 15 r = 10 + 10 r

, so

r = \frac{1}{5}

and

a = 12

.
The terms

a r^{2}, a r^{4}, a r^{6}, \dots

form a geometric progression with first term

a r^{2}

and ratio

r^{2}

, so their sum is

\frac{a r ^{2}}{1 - r ^{2}} = \frac{12 \cdot \frac{1}{25}}{\frac{24}{25}} = \frac{1}{2} .

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Single correct option