Question

You drop a ball from a height of 0.5 meter. Each curved path has 52% of the height of the previous path.

a. Write a rule for the sequence using centimeters. The initial height is given by the term n = 1.
b. What height will the ball be at the top of the third path?

Answer 1

This is a geometric sequence. The first term is the max height of the first curved path, which is 0.5. The second one is 52% of that meaning that it is 0.52 times the first term. The third term is 0.52 times the second term. Thus, in this geometric sequence,


`a = 0.5`

`r = 0.52`

You will need to use the relation
`a_n = a \cdot r^(n-1)`

Answer 2

1. 
   `f(n)=0.5(0.52)^(n-1)`
2. 0.14 m

Explanation:

The initial height of the ball is 0.5 m

Each curved path has 52% of the height of the previous path, i.e the height of the ball after one bounce will be,

`=(52)/(100)* 0.5\\\\=0.52* 0.5\ m`

The height of the ball after 2 bounces will be,

`=(52)/(100)*(0.52* 0.5)`

`=0.52*0.52* 0.5`

`=0.52^2* 0.5\ m`

Hence the series becomes,

`0.5,0.5(0.52),0.5(0.52)^2,............`

This is the case of Geometric Progression.

But as it is given that the initial height will be given by n=1, so the rules for finding the height f(n) after n bounces would be,

`f(n)=0.5(0.52)^(n-1)`

Putting n=3, we can get the height of the ball of the third path,

`\Rightarrow f(3)=0.5(0.52)^(3-1)=0.5(0.52)^(2)=0.14\ m`