Question

In Problems 51–60, write out each sum.

$\sum _{k=0}^n{\left(\frac{3}{2}\right)}^k$

Short answer

The required sum is$\sum _{k=0}^n{\left(\frac{3}{2}\right)}^k=\left\{1+\frac{3}{2}+\frac{9}{4}+......+{\left(\frac{3}{2}\right)}^n\right\}$.

Step-by-step solution

Step 1. Write the given information.

The given summation notation is:

$\sum _{k=0}^n{\left(\frac{3}{2}\right)}^k$

Step 2. Write out the sum.

To get the sum, we need to substitute $0,1,2,.....,n$for k,in the given sequence $\sum _{k=0}^n{\left(\frac{3}{2}\right)}^k$,

$$\begin{gathered} \sum _{k=0}^n{\left(\frac{3}{2}\right)}^k=\left\{{\left(\frac{3}{2}\right)}^0+{\left(\frac{3}{2}\right)}^1+{\left(\frac{3}{2}\right)}^2+......+{\left(\frac{3}{2}\right)}^n\right\} \\ =\left\{1+\frac{3}{2}+\frac{9}{4}+......+{\left(\frac{3}{2}\right)}^n\right\} \end{gathered}$$

Therefore,$\sum _{k=0}^n{\left(\frac{3}{2}\right)}^k=\left\{1+\frac{3}{2}+\frac{9}{4}+......+{\left(\frac{3}{2}\right)}^n\right\}$.