Chapter 7.4, Problem 38PS

Solution

To calculate : The upper limit on total profit a company can make over the course of its lifetime if the company had profit of $\$350,000$ in the first year. After that company’s profit has decreased by $12\%$ per year.

The upper limit on total profit a company can make over the course of its lifetime is $\$2916666.\overline{6}$ .

Given information :

The company had profit of $\$350,000$ in the first year. After that company’s profit has decreased by $12\%$ per year.

Formula used : The sum of infinite geometric series with first term $a_1$ and common ration $r$ is calculated using the formula $S=\frac{a_1}{1-r}$ .

Here, $\left|r\right|<1$ . If $\left|r\right|\ge 1$ , then the series has no sum.

Calculation :

It can be observed that company’s profit has decreased by $12\%$ per year.

The first-year company had profit of $\$350,000$ .

So,

  $a_1=350000$

On the second year, profit has decreased by $12\%$ .

So,

  $\begin{array}{c}{a}_2=\left(1-0.12\right)a_1\\ =0.88a_1\end{array}$

Similarly,

  $a_3=0.88a_2$

It can be seen that; the sequence forms a geometric sequence with value of $r$ calculated as follows.

  $\begin{array}{c}r=\frac{a_2}{a_1}\\ =\frac{0.88a_1}{a_1}\\ =0.88\end{array}$

Observe all the terms as follows.

  $\begin{array}{c}{a}_2=0.88a_1\\ =0.88\left(350000\right)\end{array}$

  $\begin{array}{c}{a}_3=0.88a_2\\ =0.88\cdot 0.88\left(350000\right)\\ =350000{\left(0.88\right)}^2\end{array}$

  $\begin{array}{c}{a}_4=0.88a_3\\ =0.88\cdot 350000{\left(0.88\right)}^2\\ =350000{\left(0.88\right)}^3\end{array}$

The general rule for the given geometric sequence is formed as $a_n=350000{\left(0.88\right)}^{n-1}$ .

For the given series the value of $a_1$ is $350000$ and $r$ is $0.88$ .

The sum of series can only exist if the common ratio is $\left|r\right|<1$ .

  $\begin{array}{c}\left|0.88\right|<1\\ 0.88<1\end{array}$

Then the sum of infinite geometric series is calculated using the formula $S=\frac{a_1}{1-r}$ .

  $\begin{array}{c}S=\frac{350000}{1-0.88}\\ =\frac{350000}{0.12}\\ =2916666.\overline{6}\end{array}$

Thus, the upper limit on total profit a company can make over the course of its lifetime is $\$2916666.\overline{6}$ .