Chapter 12, Problem 2P

Fitness Program Sheila decides to embark on a swimming program as the best way to maintain cardiovascular health. She begins by swimming 5 min on the first day, then adds $1\frac{1}{2}$ min every day after that.

1. (a) Find a recursive formula for the number of minutes T_n that she swims on the nth day of her program.
2. (b) Find the first 6 terms of the sequence T_n.
3. (c) Find a formula for T_n. What kind of sequence is this?
4. (d) On what day does Sheila attain her goal of swimming at least 65 min a day?
5. (e) What is the total amount of time she will have swum alter 30 days?

To determine

The recursive sequence that models the number of minutes $T_n$ that she swims on the nth day.

Answer to Problem 2P

The recursive sequence that models the number of minutes $T_n$ that she swims on the nth day is $T_{n-1}+\frac{3}{2}$.

Explanation of Solution

Given:

Sheila swims 5 min on the first day and $1\frac{1}{2}$ min added every day.

Formula used:

Formula to calculate the amount at the end of the nth day is,

$\left[\begin{array}{l}\text{Amount}\text{of}\text{time}\\ \text{at}\text{the}\text{end}\text{of}n\text{th}\text{day}\end{array}\right]=\left[\begin{array}{l}\text{Amount}\text{of}\text{time}\\ \text{at}\text{the}\text{end}\text{of}\text{last}\text{day}\end{array}\right]+\left[\begin{array}{l}\text{Increase}\text{in}\text{time}\\ \text{per}\text{day}\end{array}\right]$

Calculation:

Shelia swims 5 min on the first day that is $T_1=5$.

Assume $T_1$ be the number of minutes she swims at the end of first day and $T_2$ be the number of minutes she swims at the end of second day so on. The total number of minutes at the end of the nth day is $T_n$ and the time $1\frac{1}{2}$ min added every day. Thus the recursive sequence that models the number of minutes $T_n$ that she swims at the end of the nth day is given as,

$\begin{array}{c}{T}_n=T_{n-1}+1\frac{1}{2}\\ =T_{n-1}+\frac{3}{2}\end{array}$ (1)

Hence, the recursive sequence that models the number of minutes $T_n$ that she swims on the nth day is $T_{n-1}+\frac{3}{2}$.

To determine

The first six terms of the sequence $T_n$.

Answer to Problem 2P

The first six terms of the sequence $T_n$ are 5, $\frac{13}{2}$, 8, $\frac{19}{2}$, 11, and $\frac{25}{2}$.

Explanation of Solution

Given:

Sheila swims 5 min on the first day and $1\frac{1}{2}$ min added every day.

Calculation:

The value of $T_1$ is 5.

Substitute 2 for $n$ in equation (1) from part (a), to evaluate $T_2$.

$\begin{array}{c}{T}_2=\frac{3}{2}+T_1\\ =\frac{3}{2}+5\left[\because T_1=5\right]\\ =\frac{13}{2}\end{array}$

Substitute 3 for $n$ in equation (1) from part (a), to evaluate $T_3$.

$\begin{array}{c}{T}_3=\frac{3}{2}+T_2\\ =\frac{3}{2}+\frac{3}{2}+5\left[\because T_2=\frac{3}{2}+5\right]\\ =\frac{3}{2}\left(2\right)+5\\ =8\end{array}$

Substitute 4 for $n$ in equation (1) from part (a), to evaluate $T_4$.

$\begin{array}{c}{T}_4=\frac{3}{2}+T_3\\ =\frac{3}{2}+\frac{3}{2}\left(2\right)+5\left[\because T_3=\frac{3}{2}\left(2\right)+5\right]\\ =\frac{3}{2}\left(3\right)+5\\ =\frac{19}{2}\end{array}$

Substitute 5 for $n$ in equation (1) from part (a), to evaluate $T_5$.

$\begin{array}{c}{T}_5=\frac{3}{2}+T_4\\ =\frac{3}{2}+\frac{3}{2}\left(3\right)+5\left[\because T_4=\frac{3}{2}\left(3\right)+5\right]\\ =\frac{3}{2}\left(4\right)+5\\ =11\end{array}$

Substitute 6 for $n$ in equation (1) from part (a), to evaluate $T_6$.

$\begin{array}{c}{T}_6=\frac{3}{2}+T_5\\ =\frac{3}{2}+\frac{3}{2}\left(4\right)+5\left[\because T_2=\frac{3}{2}\left(4\right)+5\right]\\ =\frac{3}{2}\left(5\right)+5\\ =\frac{25}{2}\end{array}$

Hence, the first six terms of the sequence $T_n$ a are 5, $\frac{13}{2}$, 8, $\frac{19}{2}$, 11, and $\frac{25}{2}$.

To determine

The formula and behavior of sequence $T_n$.

Answer to Problem 2P

The formula of sequence $T_n$. is $\frac{3n+7}{2}$ and it is an arithmetic sequence.

Explanation of Solution

Formula used:

Formula to calculate the nth term of the arithmetic sequence $a_n$ with first term $a_1$ and common difference $d$  is,

$a_n=a_1+\left(n-1\right)d$ (2)

Given:

Sheila swims 5 min on the first day and $1\frac{1}{2}$ min added every day.

Calculation:

Since, the common difference of the sequence $T_n$ is same, the sequence $T_n$ is an arithmetic sequence.

Substitute $T_n$ for $a_n$ and 5 for $a_1$ in equation (2), to evaluate $T_n$.

$\begin{array}{c}{T}_n=5+\left(n-1\right)\frac{3}{2}\\ =5+\frac{3}{2}n-\frac{3}{2}\\ =\frac{3n+7}{2}\end{array}$ (3)

Hence, the formula of sequence $T_n$. is $\frac{3n+7}{2}$ and it is an arithmetic sequence.

To determine

The number of days when Sheila attain her goal for at least 65 min.

Answer to Problem 2P

Sheila attain her goal for at least 65 min in 41 days.

Explanation of Solution

Given:

Sheila swims 5 min on the first day and $1\frac{1}{2}$ min added every day and $T_n\ge 65$.

Calculation:

Sheila attain her goal for at least 65 min that is $T_n\ge 65$.

Substitute $65$ for $T_n$ in equation (3) from part (c), to evaluate $n$,

$\begin{array}{c}\frac{3n+7}{2}=65\\ 3n=130-7\\ n=\frac{123}{3}\\ n=41\end{array}$

Hence, Sheila attain her goal for at least 65 min in 41 days.

To determine

The total amount of time after 30 days.

Answer to Problem 2P

The total amount of time after 30 days is $48.5\min$.

Explanation of Solution

Given:

Sheila swims 5 min on the first day and $1\frac{1}{2}$ min added every day.

Calculation:

Substitute $30$ for $n$ in equation (3) from part (c), to evaluate $T_n$.

$\begin{array}{c}{T}_n=\frac{3\left(30\right)+7}{2}\\ =\frac{90+7}{2}\\ =\frac{97}{2}\\ =48.5\end{array}$

Hence, the total amount of time after 30 days is $48.5\min$.