Chapter 7.3, Problem 8GP

Solution

To calculate: The total box office revenue at U.S movie theaters in $2000$

In $1990$ , the total box office revenue at U.S movie theaters was about $\$5.02$ billion. From 1990 through 2003 the total box office revenue increased by about 5.9% per year

he total movie box office revenue for the periods $1990-2000$ was about $\$74.76$ billon

Given information:

The given terms are $a_n=5.02,n=11\text{and}\text{r}=1+0.059$ .

Formula used:

The sum nth term of a geometric sequence with first term and common ratio is given by: $s_n=\frac{a_1\left(1-r^n\right)}{1-r}$

Calculation:

Consider the given sequence.

It is given that $a_n=5.02,n=11\text{and}\text{r}=1+0.059$

The total box office revenue increased by the same percentage year after year, resulting from the geometric sequence $a_n=5.02,n=11\text{and}\text{r}=1+0.059$ .

Put the value of $a_n=5.02,n=11\text{and}\text{r}=1+0.059$ into $s_n=\frac{a_1\left(1-r^n\right)}{1-r}$

  $\begin{array}{l}{s}_n=\frac{a_1\left(1-r^n\right)}{1-r}\\ s_{14}=5.02\cdot \frac{1-{\left(1.059\right)}^{11}}{1-\left(1.059\right)}\\ s_{14}=74.76327\end{array}$

The total movie box office revenue for the periods $1990-2000$ was about $\$74.76$ billon