# Question 7a42d

Apr 6, 2017

Use the fact that repeatedly being 75% of something is the same as repeatedly multiplying by $0.75$

#### Explanation:

a)

The first bounce is:

(0.75)(100" meters") = 75" meters"

The second bounce is the first bounce multiplied by $0.75$:

(0.75)^2(100" meters") = 56.3" meters"

The third bounce is the second bounce multiplied by $0.75$:

(0.75)^3(100" meters") = 42.2" meters"#

b)
The nth bounce is:

$b \left(n\right) = {\left(0.75\right)}^{n} \left(100 \text{ meters}\right)$

The sixth bounce is:

$b \left(6\right) = {\left(0.75\right)}^{6} \left(100 \text{ meters}\right)$

$b \left(6\right) = 17.8 \text{ meters}$

Apr 6, 2017

See below.

#### Explanation:

Calling $\gamma = 0.75$ and ${h}_{0} = 100$

we have

${h}_{k} = \gamma {h}_{k - 1}$ or

${h}_{k} = {\gamma}^{k} {h}_{0}$

The three first sequence terms are

${h}_{0} , {h}_{1} , {h}_{2}$ or

${h}_{0} , {h}_{0} \gamma , {h}_{0} {\gamma}^{2} , {h}_{0} {\gamma}^{3}$ or

$100 , 75 , 56.2 , 42.2$

after the sixth bounce we have

${h}_{6} = {h}_{0} {\gamma}^{6} = 100 \times {0.75}^{6} = 17.8$[m]