Chapter 7.5, Problem 5P

Solution

To identify: The error in the student’s work if library has initially 54,000 books and each year 2% of the books are lost.

Student use the wrong rule the correct rule is $a_1=5400,a_n=0.98a_{n-1}+1150$ .

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.

The given student work is:

The student work is:

  $a_1=5400,a_n=0.02a_{n-1}+1150$

Let L be the limit of the sequence. Then:

  $\begin{array}{c}L=0.02L+1150\\ 0.98L=1150\\ L\approx 1173\end{array}$

Student use the wrong rule.

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:

  $a_1=5400,a_n=0.98a_{n-1}+1150$