Chapter 12.3, Problem 3MRE

To determine

Evaluate:

  ${\displaystyle \sum _{k=1}^{\infty }10{\left(\frac{3}{5}\right)}^{k-1}}$

Answer to Problem 3MRE

25

Explanation of Solution

Given:

  ${\displaystyle \sum _{k=1}^{\infty }10{\left(\frac{3}{5}\right)}^{k-1}}$

Calculation:

  $\begin{array}{l}{\displaystyle \sum _{k=1}^{\infty }10{\left(\frac{3}{5}\right)}^{k-1}}\\ =10\cdot \left[{\left(\frac{3}{5}\right)}^0+{\left(\frac{3}{5}\right)}^1+{\left(\frac{3}{5}\right)}^2\mathrm{...}\right]\\ =10\cdot \frac{1}{1-\frac{3}{5}}\mathrm{...................}[a+ar+a{r}^2+a{r}^3+\mathrm{...}=\frac{a}{1-r}]\\ =25\end{array}$