Question

Find $S_2:S_4$ for the series 5 + 10 + 20 + ............

Answer 1

It is given that the first term of G.P is $a_1=5$ and the second term is $a_2=10$, therefore,

$r=\frac{a_2}{a_1}=\frac{10}{5}=2$

We know that the formula for sum of $n$ terms of G.P is $S_n=\frac{a(r^n-1)}{r-1}$, where $a$ is the first term, $r$ is the common ratio.

In the given geometric series, the first term is $a=5$, the common ratio is $r=2$, therefore, consider the ratio of the sum $S_2$ and $S_4$ as follows:

$\frac{S_2}{S_4}=\frac{\frac{5(2^2-1)}{2-1}}{\frac{5(2^4-1)}{2-1}}=\frac{2^2-1}{2^4-1}=\frac{4-1}{16-1}=\frac{3}{15}=\frac{1}{5}$

Hence, $S_2:S_4=1:5$.