Chapter 11.6, Problem 3OE

To determine

To calculate:

Common ratio of given geometric series and sum series has.

Answer to Problem 3OE

Common ratio of given geometric series is $1.5$ and a sum is $-1.33$ .

Explanation of Solution

Give information:

  $\frac{2}{3}+1+\frac{3}{2}+\mathrm{...}$

Calculation:

In order to find out common ratio of given number,

  $\frac{2}{3}+1+\frac{3}{2}$

Common ratio $\left(r\right)$ is calculated as,

  $\begin{array}{l}=\frac{t_2}{t_1}\\ =\frac{1}{2/3}\\ =1.5\end{array}$

Or

  $\begin{array}{l}=\frac{3/2}{1}\\ =1.5\end{array}$

Sum of given geometric series is calculated as,

  $S=\frac{a_1}{1-r}$

Where,

  $a_1=$ Initial term.

Put the values in the equation.

  $\begin{array}{l}S=\frac{2/3}{1-\left(1.5\right)}\\ S=-1.33\end{array}$

Therefore, Common ratio of given geometric series is $1.5$ and a sum is $-1.33$ .