Answer :
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:
[tex]q=\frac{2560}{10240}=\frac{1}{4}[/tex]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:
[tex]a_n=a_{n-1}\cdot\frac{1}{4}[/tex]To write an explicit formula, we can use the model below:
[tex]a_n=a_1\cdot q^{n-1}[/tex]Since the first term is 10240, we have a1 = 10240, so the formula is:
[tex]a_n=10240\cdot(\frac{1}{4})^{n-1}[/tex]To find the height at bounce 8, let's use n = 8 and calculate the value of a8:
[tex]\begin{gathered} a_8=10240\cdot(\frac{1}{4})^{8-1} \\ a_8=10240\cdot(\frac{1}{4})^7 \\ a_8=0.625 \end{gathered}[/tex]Therefore the height after bounce 8 is equal to 0.625 cm.