Question

The Super Bounce brand of bouncy balls rebounds to 85% of the height from which it was dropped. Write both the explicit and recursive formulas h(n) that describes the height, h(n), of the bounce after n bounces given an initial height of 4ft.

Answer 1

Recursive formula
`h_n = 0.85 * h_(n-1)`

Explicit formula
`h_n = 4 * 0.85^n`

Explanation:

An explicit formula allows you to find the value of any term in the sequence. A recursive formula allows you to find the value of the nth term if you know the value of the (n-1)th term in the sequence.

Recursive formula

Given the initial height (
`h_0`), the next height (
`h_1`) is 85% of it, the following height is 85% of the new height, and so on. Mathematically:

`h_n = 0.85 * h_(n-1)`

`h_0 = 4 \; ft`

Where (
`h_n`) is the nth height, given the previous one (
`h_(n-1)`)

Explicit formula

initial height: 4 ft

n: number of bounces

h(n): height after n bounces

n h(n)

1 4*0.85 = 3.4 ft

2 3.4*0.85 = 2.89 ft

3 2.89*0.85 = 2.4565 ft

... ...

n 4*0.85^n

Answer 2

Explicit formula is
`h(n)=4(0.85)^(n-1)`.

Recursive formula is
`h_n=0.85h_(n-1)`

Explanation:

Step 1

In this step we first find the explicit formula for the height of the ball.To find the explicit formula we use the fact that the bounces form a geometric sequence. A geometric sequence has the general formula ,
`a_(n+1)=ar^(n-1).` In this case the first term
`a_o=4`, the common ratio
`r=0.85` since the ball bounces back to 0.85 of it's previous height.

We can write the explicit formula as,

`h(n)=4(0.85)^(n-1).`

Step 2

In this step we find the recursive formula for the height of the ball after each bounce. Since the ball bounces to 0.85 percent of it's previous height, we know that to get the next term in the sequence, we have to multiply the previous term by the common ratio. The general fomula for a geometric sequene is
`a_n=a_(n-1)* r.`

With the parameters given in this problem, we write the general term of the sequence as ,

`h(1)=4\\h(n)=h_(n-1)* 0.85.`