Question

Identify the following series as geometric or arithmetic. Also identify the series as infinite or finite.1/5 + 1/10 + 1/20 + 1/40 + ….Infinitearithmeticgeometricfinite

Answer 1

Given:

Given the series

$\frac{1}{5}+\frac{1}{10}+\frac{1}{20}+\frac{1}{40}+...$

Required: Identify the following series as geometric or arithmetic. Also identify the series as infinite or finite.

Explanation:

The given series is a geometric series as the terms of the series are 1/2 times its previous term.

$\begin{gathered} a_2=\frac{1}{10}=\frac{1}{2}\cdot\frac{1}{5}=\frac{1}{2}a_1 \\ a_3=\frac{1}{20}=\frac{1}{2}\cdot\frac{1}{10}=\frac{1}{2}a_2 \\ a_4=\frac{1}{40}=\frac{1}{2}\cdot\frac{1}{20}=\frac{1}{2}a_3 \end{gathered}$

So, the given series is a geometric series with first term, a = 1/5 and common difference, d = 1/2.

The three dots at the end of the series means the series is continuing. So, it is an infinite series.

Final Answer: The given series is an infinite geometric series.