Question

In Problems 61–70, express each sum using summation notation.

$a+ar+a{r}^2+.....+a{r}^{n-1}$

Short answer

The required summation notation is$a+ar+a{r}^2+.....+a{r}^{n-1}=\sum _{k=1}^{n}a{r}^{k-1}$.

Step-by-step solution

Step 1. Write the given information.

The given sum is:

$a+ar+a{r}^2+.....+a{r}^{n-1}$

Step 2. Write the sum using summation notation.

Write each term in the same format:

$$\begin{gathered} a+ar+a{r}^2+.....+a{r}^{n-1} \\ =a{\left(r\right)}^0+a{\left(r\right)}^1+a{\left(r\right)}^2+.....+a{\left(r\right)}^{n-1} \\ =a{\left(r\right)}^{1-1}+a{\left(r\right)}^{2-1}+a{\left(r\right)}^{3-1}+......+a{\left(r\right)}^{n-1} \\ =\sum _{k=1}^{n}a{r}^{k-1} \end{gathered}$$

Therefore, localid="1646799831123" $a+ar+a{r}^2+.....+a{r}^{n-1}=\sum _{k=1}^{n}a{r}^{k-1}$