The sum of the infinite G P is 15, Sum of squares of 3 terms of the g p - Maths - Sequences and Series

Question

The sum of the infinite G P is 15, Sum of squares of 3 terms of
the g p is 45 Find the series

Ankush Jain · Accepted Answer

The probable query should be:

The sum of the infinite G.P is 15 and Sum of squares of these terms is 45. Find the series.

Solution:

Let a be the first term and r be the common ratio of the infinite geometric series.

Sum of infinite g.p. = 15

$\Rightarrow \frac{a}{1 - r} = 15 ... (1)$

Also, sum of squares = 45

$\Rightarrow a^{2} + a^{2} r^{2} + a^{2} r^{4} + .... \infty = 45 \Rightarrow \frac{a^{2}}{1 - r^{2}} = 45 .... (2)$

On dividing the square of (1) by (2), we get

$\frac{\frac{a^{2}}{(1 - r)^{2}}}{\frac{a^{2}}{1 - r^{2}}} = \frac{(15)^{2}}{45} \Rightarrow \frac{(1 - r^{2})}{(1 - r)^{2}} = \frac{225}{45} \Rightarrow \frac{1 + r}{1 - r} = 5 \Rightarrow 6 r = 4 \Rightarrow r = \frac{2}{3}$

On putting value of r in (1), we get a = 5

Hence, the required series is:

$5 + \frac{10}{3} + \frac{20}{9} + ..... \infty$

The sum of the infinite G.P is 15, Sum of squares of 3 terms of the g.p is 45. Find the series.