Finding the price after 4 years means iterating the function 4 times.
$75.77
Practice makes perfect
The rate of inflation is 2 %, which means that the cost of an item in future years can be found by iterating the following function. In the formula x is the initial price.
c(x)=1.02x
We are asked to find the cost of a $70 digital audio player in 4 years. To begin, recall that iteration is a process of composing a function with itself. This process will generate a sequence in which the first term x_0 is the initial price.
x_0 ⇒ c(x_0)_(output) ⇒ c(c(x_0)_(input))
Now, if we want to find the price of the audio player in 4 years, we need to iterate the given function 4 times. We will start with the first term x_1, which is the price of the $70 audio player after 1 year.
x_1&=c( 70)
&=1.02( 70)=71.4
We can write all of the remaining terms in a table.
Year
Formula
Price
1
1.02( 70)
71.4
2
1.02( 71.4)
72.828
3
1.02( 72.828)
74.28456
4
1.02( 74.28456)
75.770251 ... ≈ 75.77
The cost of the digital audio is approximately 75.77 dollars.
