Chapter 8.3, Problem 108E

a.

To determine

To find: a geometric sequence that models the data using the exponential regression feature of a graphing utility.

Answer to Problem 108E

The geometric sequence is $a_n=26.086{(1.010471)}^n.$

Explanation of Solution

Given information: The mid-year populations $a_n$ of Peru (in millions) from 2009

through 2017 are given by ordered pairs of form $(n,a_n)$ with n =9 corresponding to 2009.

Year n	Population $a_n$
9	28.65
10	28.95
11	29.25
12	29.55
13	29.85
14	30.15
15	30.45
16	30.74
17	31.04

Formula used:

The nth term of a geometric sequence is $a_n=a_1{r}^n.$ where r is the common ratio and $a_1$ is the first term of the sequence.

Calculation:

From the given table data:

  $\begin{array}{c}{a}_9=a_1{r}^9=28.65\\ a_{10}=a_1{r}^{10}=28.95\\ \frac{a_1{r}^9}{a_1{r}^{10}}=\frac{28.65}{28.95}\\ r=\mathrm{1.010471.}\\ $ a_1{(1.01471)}^9=28.65$a_1=\mathrm{26.086.}\\ $ a_n=26.086{(1.010471)}^n.$\end{array}$

Therefore, the geometric sequence is $a_n=26.086{(1.010471)}^n.$

The graph of this geometric sequence is given below.

  

b.

To determine

To describe: the rate at which the population of Peru is growing using the sequence from part (a).

Answer to Problem 108E

The rate at which the population of Peru is growing is 1.101471.

Explanation of Solution

Given information: $a_n=26.086{(1.010471)}^n.$

Calculation:

The geometric sequence of the given population data is $a_n=26.086{(1.010471)}^n.$

Next term of the sequence is obtained by multiply the common ratio 1.101471 to the previous term so, the rate at which the population of Peru is growing is the common ratio of the geometric sequence that is 1.101471.

Therefore, the rate at which the population of Peru is growing is 1.101471.

c.

To determine

To predict: the population of Peru in 2025 using the sequence in part (a). The U.S. Census Bureau predicts that the population of Peru will be 33.28 million in 2025, how does this value compare with predicted value.

Answer to Problem 108E

The predict population of Peru in 2025 is about 33.85 million which is 0.57 million more than the U.S. Census Bureau prediction 33.28 million.

Explanation of Solution

Given information: $a_n=26.086{(1.010471)}^n.$

Calculation:

For predict the population in 2025.

Substitute n =25 into the geometric sequence equation.

  $a_{25}=26.086{(1.010471)}^{25}\approx 33.85$

Therefore, the predict population of Peru in 2025 is about 33.85 million which is 0.57 million more than the U.S. Census Bureau prediction 33.28 million.

d.

To determine

To find: when the population of Peru will reach 39 million.

Answer to Problem 108E

The population of Peru will reach 39 million in 2037.

Explanation of Solution

Given information: $a_n=26.086{(1.010471)}^n.$

Calculation:

To find when the population of Peru will reach 39 million.

  $\begin{array}{c}$ a_n=26.086{(1.010471)}^n.$$ 39=26.086{(1.010471)}^n.$$ {(1.010471)}^n=\frac{39}{26.086}=1.49505$Taking\log ofbothsides.\\ n\log (1.010471)=\log (1.49505)\\ n=\frac{\log (1.49505)}{\log (1.010471)}\approx 38.61\approx 37\end{array}$

Therefore, the population of Peru will reach 39 million in 2037.