Exercise 60 Page 449

Practice makes perfect

a Let

a_{n}

denote the number of books in the library at the beginning of the

n^{th}

year. The library initially has

54000

books. This tells us that

a_{1} = 54000 .

Next, we will find a recursive equation for the sequence

a_{n} .

Recursive Equation

At the beginning of the

(n - 1)^{th}

year the number of books is

a_{n - 1} .

During this year,

2 %

of the books are lost or discarded. Therefore,

98 % = 0.98

of the old books remain. Furthermore,

1150

new books are added to the library.

Books at the start of the n^{th} year ⇓ a_{n} = 0.98 \cdot = 0.98 \cdot Books from previous year ⇓ a_{n - 1} + + New books ⇓ 1150

Therefore, we can write a recursive rule:

a_{1} = 54000,

a_{n} = 0.98 a_{n - 1} + 1150

for

n > 1 .

b We will graph the sequence from Part A using a calculator. First, we push

MODE

and change Func to Seq.

Now we are dealing with sequences in our calculator. Push $Y =,$ and we get the following.

If we want to enter a recursive sequence we enter $u (n - 1)$ and $u (n - 2)$ for the previous terms. Enter $0.98 * u (n - 1)$ into the row with $⋱ u (n) .$ The initial value or values we can enter in the row with $u (nMin) .$ Now, we will input the given recursive rule of the sequence from Part A into the calculator.

Now, to draw the sequence we push $GRAPH .$

By pressing the right arrow we can find that the values of the sequence stabilize around $57500 .$ This tells us that the number of books in the library stabilizes around $57500$ over time.

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Recursive Equation