Chapter 3.6, Problem 31PPS

a.

To determine

To determine the formula for the nth term.

Answer to Problem 31PPS

  $a_n=2.5n-0.5$

Explanation of Solution

Given:

Sequence:

  $3,3.5,4,4.5,\mathrm{5...}$

Formula used:

nth term of any arithmetic progression is given by the formula:

  $t_n=a+(n-1)d$

  $t_n=$ nth term

  $a=$ The first term of the arithmetic progression

  $d=$ The common difference of the arithmetic progression

Calculation / Explanation:

The common difference of the arithmetic progression is:

  $3.5-3=0.5$

Applying above formula,

  $\Rightarrow a_n=a+(n-1)d$

  $\Rightarrow a_n=3+(n-1)(0.5)$  $[\because a=3\&d=0.5]$

  $\begin{array}{l}\Rightarrow a_n=3+0.5n-0.5\\ \Rightarrow a_n=2.5n-0.5\end{array}$

Conclusion:

The required nth term formula of the sequence is $a_n=2.5n-0.5$ .

b.

To determine

To determine the week in which she will run $10$ miles per day.

Answer to Problem 31PPS

15^th

Explanation of Solution

Given:

Sequence:

  $3,3.5,4,4.5,\mathrm{5...}$

Formula used:

nth term of any arithmetic progression is given by the formula:

  $t_n=a+(n-1)d$

  $t_n=$ nth term

  $a=$ The first term of the arithmetic progression

  $d=$ The common difference of the arithmetic progression

Calculation:

The common difference of the arithmetic progression ia:

  $-1-(-3)=-1+3=2$

Let, she will run $10$ miles per day on nth week.

Applying above formula,

  $\Rightarrow a_n=a+(n-1)d$

  $\Rightarrow a_n=3+(n-1)(0.5)$  $[\because a=3\&d=0.5]$

  $\begin{array}{l}\Rightarrow 10=3+0.5n-0.5[a_n=10]\\ \Rightarrow 10-3+0.5=0.5n\\ \Rightarrow 7.5=0.5n\\ \Rightarrow n=15\end{array}$

Conclusion:

Therefore, she will run $10$ miles per day on15th week.

c.

To determine

To justify the statement.

Answer to Problem 31PPS

No, this pattern will not continue infinitely.

Explanation of Solution

Given:

Olivia plans to run three miles per day for the first week.

Then she will increase daily distance by a half mile each of the following weeks.

Calculation:

In some bigger cases the number of miles ran per day will become unrealistic.

Conclusion:

Therefore, Olivia cannot continue this patter infinitely.