Chapter 14

Find the sum of the first 30 terms of the arithmetic sequence \(0,2,4,6,8, \ldots .870\)

Suppose that your savings account contains \(\$ 3750\) at the beginning of a year. If you withdraw \(\$ 250\) per month from the account, how much will it contain at the end of the year?

Find the sum of the first 40 terms of the arithmetic sequence \(2,6,10,14,18, \ldots\) 3200

Solve each equation for the indicated variable. $$ 2 x-5 y=7 \text { for } x $$

Sonya decides to start saving dimes. She plans to save 1 dime the first day of April, 2 dimes the second day, 3 dimes the third day, 4 dimes the fourth day, and so on for the 30 days of April. How much money will she save in April? \(\$ 46.50\)

Find the sum of the first 60 terms of the arithmetic sequence \(-2,3,8,13,18, \ldots 8730\)

$$ 5 x-6 y=12 \text { for } x $$

Nancy decides to start saving dimes. She plans to save 1 dime the first day of April, 2 dimes the second day, 4 dimes the third day, 8 dimes the fourth day, and so on for the first 15 days of April. How much will she save in 15 days? \(\$ 3276.70\)

Find the sum of the first 80 terms of the arithmetic sequence \(7,3,-1,-5,-9, \ldots\). \(-12,080\)

$$ -7 x-y=4 \text { for } y $$

A tank contains 61,440 gallons of water. Each day onefourth of the water is drained out. How much water remains in the tank at the end of 6 days? 10,935 gallons

$$ \sum_{i=1}^{6} 3^{i} 1092 $$

$$ 3 x-2 y=-1 \quad \text { for } y $$

Show a mathematical induction proof. Prove that \(5^{n}>5 n-1\) for all positive integer values of \(n\).

$$ \sum_{i=2}^{5}(-3)^{i+1} 540 $$

Find the sum of the first 50 terms of the arithmetic sequence \(\frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, \ldots .637 .5\)

$$ 3(x-2 y)=4 \quad \text { for } x $$

Prove that \(n^{3}-n+3\) is divisible by 3 for all positive integer values of \(n\).

$$ \sum_{i=1}^{6} 3\left(\frac{1}{2}\right)^{i} \quad 2 \frac{61}{64} $$

Find the indicated sum. 1+5+9+13+\cdots+197 \quad 4950

$$ \frac{y-a}{b}=\frac{x+b}{c} \text { for } x $$

$$ \sum_{i=1}^{5} 2\left(\frac{1}{3}\right)^{i} \frac{242}{243} $$

$$ \frac{x-a}{b}=\frac{y-a}{c} \text { for } y $$

Find the sum of each infinite geometric sequence. If the sequence has no sum, so state. \(2,1, \frac{1}{2}, \frac{1}{4}, \ldots\) 4

2+8+14+20+\cdots+146 \quad 1850

$$ (y+1)(a-3)=x-2 \text { for } y $$

6+9+12+15+\cdots+93

$$ (y-2)(a+1)=x \quad \text { for } y $$

(-7)+(-10)+(-13)+(-16)+\cdots+(-109) \quad-2030 \quad=

Solve each of Problems 47– 62 by setting up and solving an appropriate algebraic equation. Suppose that the length of a certain rectangle is \(2 \mathrm{me}-\) ters less than four times its width. The perimeter of the rectangle is 56 meters. Find the length and width of the rectangle.

$$ (-5)+(-9)+(-13)+(-17)+\cdots+(-169) $$ \(-3654\)

The perimeter of a triangle is 42 inches. The second ide is 1 inch more than twice the first side, and the hird side is 1 inch less than three times the first side. ind the lengths of the three sides of the triangle.

How long will it take \(\$ 500\) to double itself at \(9 \%\) simple interest?

How long will it take \(\$ 700\) to triple itself at \(10 \%\) simple interest?

Find the sum of the first 200 odd whole numbers. 40,000

How long will it take \(P\) dollars to double itself at \(9 \%\) simple interest?

$$ 2,-6,18,-54, \ldots $$ No sum

Find the sum of all even numbers between 18 and 482 , inclusive. 58,250

How long will it take \(P\) dollars to triple itself at \(10 \%\) simple interest?

\frac{1}{2}, \frac{3}{8}, \frac{9}{32}, \frac{27}{128}, \ldots .2

Find the sum of all even numbers between 18 and 482 , inclusive. 58,250

Two airplanes leave Chicago at the same time and fly in opposite directions. If one travels at 450 miles per hour ind the other at 550 miles per hour, how long will it ake for them to be 4000 miles apart?

\(4,-\frac{4}{3}, \frac{4}{9},-\frac{4}{27}, \ldots\) 3

Find the sum of all odd numbers between 17 and 379 , inclusive. 36,036

Find the sum of the first 30 terms of the arithmetic sequence with the general term \(a_{n}=5 n-4 . \quad 2205\)

Juan starts walking at 4 miles per hour. An hour and a half later, Cathy starts jogging along the same route at 6 miles per hour. How long will it take Cathy to catch up with Juan?

Find the sum of the first 40 terms of the arithmetic sequence with the general term \(a_{n}=4 n-7 . \quad 3000\)

A car leaves a town at 60 kilometers per hour. How long will it take a second car, traveling at 75 kilometers per hour, to catch the first car if it leaves 1 hour later?

For Problems 57– 68, change each repeating decimal to ab form, where a and b are integers and b 0. Express ab in reduced form. 0 . \overline{3} \quad \frac{1}{3}

Find the sum of the first 25 terms of the arithmetic sequence with the general term \(a_{n}=-4 n-1\). \(-1325\)