Chapter 8.3, Problem 60E

To determine

To find: the sum and verify the result by graphing utility.

Answer to Problem 60E

  ${\displaystyle \sum _{n=0}^{15}10{\left(\frac{7}{6}\right)}^n}=646.8$

Explanation of Solution

Given information:

Given sum

  ${\displaystyle \sum _{n=0}^{15}10{\left(\frac{7}{6}\right)}^n}$

Formula used:

The sum $S_n$ or ${\displaystyle \sum _{i=1}^n{a}_r{r}^{r-1}}$ of a finite geometric sequence is $a_1\left(\frac{1-r^n}{1-r}\right)$ , where $a_1$ is the first term, $r$ is the common ration, and $n$ is the number of terms in the sequence.

Calculation:

We know that the index of a geometric series with $n=1$ . If the index starts with 0, then adjust the formula for the $nth$ partial sum.

  $\begin{array}{l}{\displaystyle \sum _{n=0}^{15}10{\left(\frac{7}{6}\right)}^n}=10{\left(\frac{7}{6}\right)}^0+{\displaystyle \sum _{n=1}^{15}10{\left(\frac{7}{6}\right)}^n}\\ {\displaystyle \sum _{n=0}^{15}10{\left(\frac{7}{6}\right)}^n}=10+{\displaystyle \sum _{n=1}^{15}10{\left(\frac{7}{6}\right)}^n}\end{array}$

Now,

  ${\displaystyle \sum _{n=1}^{15}10{\left(\frac{7}{6}\right)}^n}=10{\left(\frac{7}{6}\right)}^1+\mathrm{....}+10{\left(\frac{7}{6}\right)}^{15}$

Substitute 10for $a_1$ , $\frac{7}{6}$ for $r_1$ ,and 15 for $n$ in $S_n=a_1\left(\frac{r^n-1}{r-1}\right)\text{as}r>1$

  $\begin{array}{c}{S}_{15}=a_1\left(\frac{r^{20}-1}{r-1}\right)\\ =10\left(\frac{{\left(\frac{7}{6}\right)}^{15}-1}{\frac{7}{6}-1}\right)\\ =636.805441683119\\ \approx 636.8\end{array}$

Replace ${\displaystyle \sum _{n=1}^{15}10{\left(\frac{7}{6}\right)}^n}$ with 2688.31 in ${\displaystyle \sum _{n=0}^{15}10{\left(\frac{7}{6}\right)}^n}=10+{\displaystyle \sum _{n=1}^{15}10{\left(\frac{7}{6}\right)}^n}$

  $\begin{array}{c}{\displaystyle \sum _{n=0}^{15}10{\left(\frac{7}{6}\right)}^n}=10+{\displaystyle \sum _{n=1}^{15}10{\left(\frac{7}{6}\right)}^n}\\ =10+636.8\\ =646.8\end{array}$

Thus, the sum is 646.8.

Verify the result using graphing utility as shown:

1. Choose the sum feature from the math menu of the list feature.
2. Choose the sequence feature from the operations menu of the list feature.
3. Enter the expression for the sequence, the variable, the lower limit of summation, and the upper limit of summation. Press ENTER to get the sum as $646.8$ .