Chapter 7.5, Problem 4P

Solution

To identify: What happens to the number of books over the time if library has initially 54,000 books and each year 2% of the books are lost. The library can afford to purchase 1150 new books each year. Use the graphing method and algebraic method.

The number of books over the time approaches to $57500$ .

Given information:

Library has initially 54,000 books and each year 2% of the books are lost. The library can afford to purchase 1150 new books each year.

Explanation:

Consider the given information.

library has initially 54,000 books and each year 2% of the books are lost. The library can afford to purchase 1150 new books each year.

Let they have 100% books at the starting then after losing 2% books, they have 98% books left and 1150 new books added each year.

So, the rule can be written as:

  $a_1=5400,a_n=0.98a_{n-1}+1150$

Assume that the sequence has a limit L , which is the value that the sequence approaches as n becomes large.

As n become larger the left-hand side approaches to L while the value of the right-hand side approaches to $0.98L+1150$ .

  $\begin{array}{c}L=0.98L+1150\\ 0.02L=1150\\ L=57500\end{array}$

The sequence approaches to the limit $L=57500$ .

Now solve this graphically.

Step 1: Open Ti-83 calculator press MODE key select Seq and Dot option.

  

Step 2: Press Y= key and enter the given sequence. Press the key $\text{X,T,θ,n}$ to write n .

  

Step 3: Press GRAPH key.

  

The sequence approaches to the limit $L=57500$ .