Chapter 7.3, Problem 16Q

Solution

To calculate : The rule for average tuition $a_n$ in terms of the year and average tuition at public college in 2022 if in 1995, the average tuition was $\$2057$ . From 1995 to 2002, the average tuition at public colleges increased about $6\%$ per year.

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

Given information :

In 1995, the average tuition was $\$2057$ . From 1995 to 2002, the average tuition at public colleges increased about $6\%$ per year.

Formula used :

The $n^{th}$ term of a geometric series with first term as $a_1$ and common ratio $r$ is given by $a_n=a_1{r}^{n-1}$ .

Calculation :

It can be observed that average tuition at public colleges increased about $6\%$ per year.

The first-year tuition fees were $\$2057$ .

So,

  $a_1=2057$

On the second year, fees have increased by $6\%$ .

So,

  $\begin{array}{c}{a}_2=\left(1+0.06\right)a_1\\ =1.06a_1\\ =2057\left(1.06\right)\end{array}$

Similarly,

  $\begin{array}{c}{a}_3=1.06a_2\\ =1.06\cdot 2057\left(1.06\right)\\ =2057{\left(1.06\right)}^2\end{array}$

The general rule for the given geometric sequence is formed as $a_n=2057{\left(1.06\right)}^{n-1}$ .

For the given series the value of $a_1$ is $2057$ and $r$ is $1.06$ .

Till 2002, there will be $\left(2002-1995\right)+1$ years, so $n=8$ .

So,

  $\begin{array}{c}{a}_8=2057{\left(1.06\right)}^{8-1}\\ =2057{\left(1.06\right)}^7\\ =2057\left(1.5036\right)\\ =3092.97\end{array}$

  $a_8\approx \$3093$

Thus, rule for average tuition $a_n$ in terms of the year is $a_n=2057{\left(1.06\right)}^{n-1}$ and average tuition at public college in 2022 is $\$3093$ .