Chapter 11.2, Problem 70PPS

(a)

To determine

Write an equation to represent the nth term of the sequence.

Answer to Problem 70PPS

  $a_n=2.5-0.5n$

Explanation of Solution

Given:

While training for cross country, Silvia plans to run 3 miles per day for the first week, and then increase the distance by a half miles each of the following weeks.

Concept Used:

According to the question, the information is given: $a_1=3;d=0.5;\text{find}{a}_n$

Calculation:

According to the question, the information is given: $a_1=3;d=0.5;\text{find}{a}_n$

  $\begin{array}{l}{a}_1=3;d=0.5;\text{find}{a}_n\\ a_n=a_1+(n-1)·d\\ a_n=3+(n-1)·0.5\\ a_n=3+0.5n-0.5\\ a_n=2.5-0.5n\end{array}$

Thus, the nth term of the series is $a_n=2.5-0.5n$

(b)

To determine

Find the number of week when she will be running 10 miles per day.

Answer to Problem 70PPS

15^th week

Explanation of Solution

Given:

While training for cross country, Silvia plans to run 3 miles per day for the first week, and then increase the distance by a half miles each of the following weeks.

Concept Used:

The nth term of the series is $a_n=2.5-0.5n$

Calculation:

When $a_n=10$ , find the number of week’s n from the nth term of the series $a_n=2.5-0.5n$

  $\begin{array}{l}{a}_n=10;\text{find }n\\ a_n=2.5-0.5n\\ 10=2.5-0.5n\\ 10-2.5=2.5-2.5-0.5n\\ 7.5=0.5n\\ \frac{7.5}{2.5}=\frac{0.5n}{0.5}\\ n=15\end{array}$

Thus, on 15^th week she will be running 10 miles per day.

(c)

To determine

It is reasonable for this pattern to continue indefinitely.

Answer to Problem 70PPS

Explanation of Solution

Given:

While training for cross country, Silvia plans to run 3 miles per day for the first week, and then increase the distance by a half miles each of the following weeks.

Concept Used:

Is it reasonable for this pattern to continue indefinitely?

No; eventually the number of miles per day will become unrealistic.