Question

After each bounce, the height of the ball is measured and recorded in the table.Bounce NumberHeight (cm) After Bounce110,24022,56036404160Write an explicit and recursive formula to represent the sequence of the height of the ball after each bounce. What is the height of the ball after bounce 8? Explain your answer by using one of the formulas you created.

Answer 1

Looking at the height values, we can see that the values are being multiplied by a constant value (we have a geometric sequence).

To find the common ratio, let's divide one term by the term before.

Using bounces 2 and 1, we have:

`q=(2560)/(10240)=(1)/(4)`

The recursive formula relates one element of the sequence with the previous element,

Since each value in the table is 4 times less than the value before, we can write the recursive formula below:

`a_n=a_(n-1)\cdot(1)/(4)`

To write an explicit formula, we can use the model below:

`a_n=a_1\cdot q^(n-1)`

Since the first term is 10240, we have a1 = 10240, so the formula is:

`a_n=10240\cdot((1)/(4))^(n-1)`

To find the height at bounce 8, let's use n = 8 and calculate the value of a8:

`\begin{gathered} a_8=10240\cdot((1)/(4))^(8-1) \\ a_8=10240\cdot((1)/(4))^7 \\ a_8=0.625 \end{gathered}`

Therefore the height after bounce 8 is equal to 0.625 cm.