Question

Find the sum of the infinite geometric series 1+ 0.2+0.04 +0.008 +OA. 1.25OB.2OC the series divergesOD.3Reset Selection

Answer 1

Given: The series below

$1+0.2+0.04+0.008+...$

To Determine: The sum of the infinite geometric series

Solution

Let us determine the common ratio

For a geometric series with terms a1, a2, a3, ... as below

$\begin{gathered} a_1+a_2+a_3+...+a_{n-1}+a_n \\ r=\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_n}{a_{n-1}} \end{gathered}$

Let us apply the above to the given question

$\begin{gathered} ratio(r)=\frac{0.2}{1}=\frac{0.04}{0.2}=\frac{0.008}{0.04}=0.2 \\ r=0.2 \end{gathered}$

The first term a is 1

$a=1$

Please note the below

Since the common ratio r is less than 1, the series converges, the sum to inifinity would be

$\begin{gathered} S_{\infty}=\frac{a}{1-r} \\ a=1 \\ r=0.2 \\ S_{\infty}=\frac{1}{1-0.2} \\ S_{\infty}=\frac{1}{0.8} \\ S_{\infty}=\frac{10}{8} \\ S_{\infty}=1.25 \end{gathered}$

Hence, the sum of the infinite geometric series is 1.25, OPTION A