Chapter 14

Bret started on a 70 -mile bicycle ride at 20 miles per hour. After a time he became a little tired and slowed down to 12 miles per hour for the rest of the trip. The entire trip of 70 miles took \(4 \frac{1}{2}\) hours. How far had Bret ridden when he reduced his speed to 12 miles per hour?

How many gallons of a \(12 \%-\) salt solution must be mixed with 6 gallons of a \(20 \%\)-salt solution to obtain a \(15 \%\)-salt solution?

$$ \sum_{i=1}^{45}(5 i+2) \quad 5265 $$

Suppose that you have a supply of a \(30 \%\) solution of alcohol and a \(70 \%\) solution of alcohol. How many quarts of each should be mixed to produce 20 quarts that is \(40 \%\) alcohol?

How many cups of grapefruit juice must be added to 40 cups of punch that is \(5 \%\) grapefruit juice to obtain a punch that is \(10 \%\) grapefruit juice?

0 . \overline{123} \quad \frac{41}{333}

$$ \sum_{=1}^{30}(-2 i+4)-810 $$

A 16-quart radiator contains a \(50 \%\) solution of antifreeze. How much needs to be drained out and replaced with pure antifreeze to obtain a \(60 \%\) antifreeze solution?

$$ 0 . \overline{273} \quad \frac{91}{333} $$

$$ \sum_{i=1}^{40}(-3 i+3)-2340 $$

A 16-quart radiator contains a \(50 \%\) solution of antifreeze. How much needs to be drained out and replaced with pure antifreeze to obtain a \(60 \%\) antifreeze solution?

$$ \sum_{i=4}^{32}(3 i-10) $$ 1276

Some people subtract 32 and then divide by 2 to estimate the change from a Fahrenheit reading to a Celsius reading. Why does this give an estimate and how good is the estimate?

\sum_{i=6}^{7}(4 i-9) \quad 4074

\sum_{i=10}^{20} 4 i \quad 660

Solve \(i=P r t\) for \(i\), given that \(P=\$ 875, r=12 \frac{1}{2} \%\), and \(t=4\) years.

\sum_{i=1}^{5} i^{2} 55

\sum_{-1}^{6}\left(i^{2}+1\right) 97

Solve \(i=P r t\) for \(t\), given that \(i=\$ 453.25, P=\$ 925\), and \(r=14 \%\).

Explain the difference between an arithmetic sequence and a geometric sequence.

\sum_{i=3}^{8}\left(2 i^{2}+i\right) \quad 431

What does it mean to say that the sum of the infinite geometric sequence \(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots\) is \(2 ?\)

\sum_{i=4}^{7}\left(3 i^{2}-2\right) \quad 370

What do we mean when we say that the infinite geometric sequence \(1,2,4,8, \ldots\) has no sum?

Before developing the formula \(a_{n}=a_{1}+(n-1) d\), we stated the equation \(a_{k+1}-a_{k}=d\). In your own words, explain what this equation says.

Solve \(i=\) Prt for \(r\), given that \(i=\$ 159.50, P=\$ 2200\), and \(t=0.5\) of a year. Express \(r\) as a percent.

Why don't we discuss the sum of an infinite arithmetic sequence?

Explain how to find the sum \(1+2+3+4+\cdots+175\) without using the sum formula.

Explain in words how to find the sum of the first \(n\) terms of an arithmetic sequence.

Solve \(A=P+P r\) for \(P\), given that \(A=\$ 2173.75\), \(r=8 \frac{3}{4} \%\), and \(t=2\) years.

Explain how one can tell that a particular sequence is an arithmetic sequence.

1–8, Express the given inequality in interval notation and sketch a graph of the interval. x>1

$$ a_{n}= \begin{cases}\frac{1}{n} & \text { for } n \text { odd } \\ n^{2} & \text { for } n \text { cven } \\ 1,4, \frac{1}{3}, 16, \frac{1}{5}, 36\end{cases} $$

x \geq-1

$$ I_{n}= \begin{cases}5 n-1 & \text { for } n \text { a multiple of } 3 \\ 2 n & \text { otherwise }\end{cases} $$

Write the first six terms of each sequence. $$ \left\\{\begin{array}{l} a_{1}=4 \\ a_{n}=3 a_{n-1} \quad \text { for } n \geq 2 \end{array} \quad 4,12,36,108,324,972\right. $$

$$ \left\\{\begin{array}{l} a_{1}=3 \\ a_{n}=a_{n-1}+2 \quad \text { for } n \geq 2 \end{array} 5,7,9,11,13\right. $$

x<1

$$ \left\\{\begin{array}{l} a_{1}=1 \\ a_{2}=1 \\ a_{n}=a_{n-2}+a_{n-1} \quad \text { for } n \geq 3 \end{array} \quad 1,1,2,3,5,8\right. $$

$$ \left\\{\begin{array}{l} a_{1}=2 \\ a_{2}=3 \\ a_{n}=2 a_{n-2}+3 a_{n-1} \quad \text { for } n \geq 3 \end{array} \quad 2,3,13,45,161,573\right. $$

\left\\{\begin{array}{l} a_{1}=3 \\ a_{2}=1 \\ a_{n}=\left(a_{n-1}-a_{n-2}\right)^{2} \quad \text { for } n \geq 3 \end{array} \quad 3,1,4,9,25,256\right.

\left\\{\begin{array}{l} a_{1}=1 \\ a_{2}=2 \\ a_{3}=3 \\ a_{n}=a_{n-1}+a_{n-2}+a_{n-3} \quad \text { for } n \geq 4 \end{array} \quad 1,2,3,6,11,20\right.