Chapter 11.5, Problem 61STP

(a)

To determine

Write the first five terms of a sequence describing his training schedule.

Answer to Problem 61STP

  $2,3,4.5,6.75,10.125$

Explanation of Solution

Given:

Adrahan is training for a marathon, about 26 miles. He begins by running 2 miles. Then, when he runs every other day, he runs one and a half times the distance he ran the time before.

Write the first five terms of a sequence describing his training schedule.

Concept Used:

This forms a geometric sequence. Given: $a_1=2;\text{and}r=1.5$

Calculation:

Given: $a_1=2;\text{and}r=1.5$

  $\begin{array}{l}{a}_n=a_{n-1}·r\\ a_2=a_1·r=2·1.5=3\\ a_3=a_2·r=3·1.5=4.5\\ a_4=a_3·r=4.5·1.5=6.75\\ a_5=a_4·r=6.75·1.5=10.125\end{array}$

  $2,3,4.5,6.75,10.125$

Thus, the five terms are $2,3,4.5,6.75,10.125$

(b)

To determine

Find the time when he will exceed 26 miles in one run?

Answer to Problem 61STP

He will exceed 26 miles in one run on the eighth session.

Explanation of Solution

Given:

Adrahan is training for a marathon, about 26 miles. He begins by running 2 miles. Then, when he runs every other day, he runs one and a half times the distance he ran the time before.

When will he exceed 26 miles in one run?

Concept Used:

Given: $a_n=26$

Calculation:

Given: $a_n=26$

  $\begin{array}{l}{a}_n=26;\text{Find}n\\ a_n=a_1{r}^{n-1}\\ $ 26=2·{(1.5)}^{n-1}${1.5}^{n-1}=13\Rightarrow n=7.3\end{array}$

He will exceed 26 miles in one run on the eighth session.

Thus, he will exceed 26 miles in one run on the eighth session.

(c)

To determine

Find the time when he will have run 100 total miles.

Answer to Problem 61STP

He will have run 100 total miles during the ninth session.

Explanation of Solution

Given:

Adrahan is training for a marathon, about 26 miles. He begins by running 2 miles. Then, when he runs every other day, he runs one and a half times the distance he ran the time before.

When will he have run 100 total miles?

Concept Used:

Given: ${\text{S}}_n=100$ ; ${\text{S}}_n=100;\text{Find}n$

Calculation:

Given: ${\text{S}}_n=100$

  $\begin{array}{l}{\text{S}}_n=100;\text{Find}n\\ {\text{S}}_n=\frac{a_1(r^n-1)}{r-1}\text{when}r>1\\ 100=\frac{2({1.5}^n-1)}{1.5-1}\\ 25=({1.5}^n-1)\\ {1.5}^n=26\Rightarrow n=8.04\end{array}$

He will have run 100 total miles during the ninth session.

Thus, He will have run 100 total miles during the ninth session.