Chapter 12

Determine the positive integer values of \(n\), where the arithmetic sequence \(a_{n}\) satisfies the inequality. $$a_{n}<22, \text { where } a_{n}=2 n-1$$

Decide whether the situation described involves a permutation or a combination of objects. (a) A telephone number (b) A Social Security number (c) A hand of cards in poker (d) A committee of politicians (e) The "combination" on a combination lock (f) A lottery choice of six numbers where the order does not matter (g) An automobile license plate

Write the binomial expansion for each expression. $$(3 x-2 y)^{6}$$

Use a formula to find the sum of each series. $$\sum_{j=1}^{6} 48\left(\frac{1}{2}\right)^{j}$$

Gender Makeup of a Family Suppose a family has 5 children. Suppose also that the probability of having a girl is \(\frac{1}{2} .\) Find the probability that the family has the following children. (picture cannot copy) No boys

Prove each statement by mathematical induction. \(4^{n}>n^{4},\) for \(n \geq 5\)

Determine the positive integer values of \(n\), where the arithmetic sequence \(a_{n}\) satisfies the inequality. $$a_{n}>0, \text { where } a_{n}=5-n$$

Explain the difference between a permutation and a combination.

Write the binomial expansion for each expression. $$(7 k-9 j)^{4}$$

Use a formula to find the sum of each series. $$\sum_{j=1}^{5} 243\left(\frac{2}{3}\right)^{j}$$

Find the sum for each series. $$\sum_{i=1}^{5} \frac{1}{i+1}$$

Gender Makeup of a Family Suppose a family has 5 children. Suppose also that the probability of having a girl is \(\frac{1}{2} .\) Find the probability that the family has the following children. (picture cannot copy) At least 3 boys

Prove each statement by mathematical induction. $$\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right) \cdots\left(1-\frac{1}{n^{2}}\right)=\frac{n+1}{2 n}, \text { if } n \geq 2$$

Determine the positive integer values of \(n\), where the arithmetic sequence \(a_{n}\) satisfies the inequality. $$a_{n} \geq 2, \text { where } a_{n}=5-\frac{1}{2} n$$

Use the fundamental principle of counting or permutations to solve each problem. How many different types of homes are available if a builder offers a choice of 5 basic plans, 3 roof styles, and 2 exterior finishes?

Write the binomial expansion for each expression. $$\left(\frac{m}{2}-1\right)^{6}$$

Use a formula to find the sum of each series. $$\sum_{k=4}^{10}(-2)^{k}$$

Find the sum for each series. $$\sum_{i=1}^{4} i^{i}$$

Gender Makeup of a Family Suppose a family has 5 children. Suppose also that the probability of having a girl is \(\frac{1}{2} .\) Find the probability that the family has the following children. (picture cannot copy) No more than 4 girls

Prove each statement by mathematical induction. \(4 n<2^{n},\) if \(n \geq 5\)

Determine the positive integer values of \(n\), where the arithmetic sequence \(a_{n}\) satisfies the inequality. $$a_{n} \geq 30, \text { where } a_{n}=100-2 n$$

Use the fundamental principle of counting or permutations to solve each problem. An auto manufacturer produces 7 models, each available in 6 different colors, 4 different upholstery fabrics, and 5 interior colors. How many varieties of the auto are available?

Write the binomial expansion for each expression. $$\left(3+\frac{y}{3}\right)^{5}$$

Use a formula to find the sum of each series. $$\sum_{k=3}^{9}(-3)^{k}$$

Find the sum for each series. $$\sum_{i=1}^{5} i^{i-1}$$

The table gives the results of a survey of \(14,000\) college students who were cigarette smokers in a recent year. $$\begin{array}{|l|c|} \hline \begin{array}{l} \text { Number of Cigarettes } \\\ \text { per Day } \end{array} & \begin{array}{c} \text { Percent } \\\ \text { (as a decimal) } \end{array} \\ \hline \text { Less than } 1 & 0.45 \\\ 1 \text { to } 9 & 0.24 \\ 10 \text { to } 19 & 0.20 \\ \text { A pack of } 20 \text { or more } & 0.11 \end{array}$$ Using the percents as probabilinies, approximate the probability that, out of 10 of these shudent smokers selected at random, the following were true. Four smoked fewer than 10 cigarettes per day.

Solve each problem. Number of Handshakes Suppose that each of the \(n\) \((n \geq 2)\) people in a room shakes hands with everyone else, but not with himself. Show that the number of handshakes is \(\frac{n^{2}-n}{2}\)

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{1}=8, d=3$$

Use the fundamental principle of counting or permutations to solve each problem. How many different 4-letter radio station call letters can be made (a) if the first letter must be \(\mathrm{K}\) or \(\mathrm{W}\) and no letter may be repeated? (b) if repeats are allowed (but the first letter is \(\mathrm{K}\) or \(\mathrm{W}\) )? (c) How many of the 4 -letter call letters (starting with K or W) with no repeats end in R?

Write the binomial expansion for each expression. $$\left(\sqrt{2} r+\frac{1}{m}\right)^{4}$$

Use a formula to find the sum of each series. $$\sum_{i=2}^{8}-2^{i}$$

Find the sum for each series. $$\sum_{k=1}^{6}(-1)^{k} \cdot k$$

The table gives the results of a survey of \(14,000\) college students who were cigarette smokers in a recent year. $$\begin{array}{|l|c|} \hline \begin{array}{l} \text { Number of Cigarettes } \\\ \text { per Day } \end{array} & \begin{array}{c} \text { Percent } \\\ \text { (as a decimal) } \end{array} \\ \hline \text { Less than } 1 & 0.45 \\\ 1 \text { to } 9 & 0.24 \\ 10 \text { to } 19 & 0.20 \\ \text { A pack of } 20 \text { or more } & 0.11 \end{array}$$ Using the percents as probabilinies, approximate the probability that, out of 10 of these shudent smokers selected at random, the following were true. Five smoked a pack or more per day.

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{1}=-9, d=4$$

Write the binomial expansion for each expression. $$\left(\frac{1}{k}-\sqrt{3} p\right)^{3}$$

Find the sum for each series. $$\sum_{i=1}^{7}(-1)^{i+1} \cdot i^{2}$$

The table gives the results of a survey of \(14,000\) college students who were cigarette smokers in a recent year. $$\begin{array}{|l|c|} \hline \begin{array}{l} \text { Number of Cigarettes } \\\ \text { per Day } \end{array} & \begin{array}{c} \text { Percent } \\\ \text { (as a decimal) } \end{array} \\ \hline \text { Less than } 1 & 0.45 \\\ 1 \text { to } 9 & 0.24 \\ 10 \text { to } 19 & 0.20 \\ \text { A pack of } 20 \text { or more } & 0.11 \end{array}$$ Using the percents as probabilinies, approximate the probability that, out of 10 of these shudent smokers selected at random, the following were true. Fewer than 2 smoked between 1 and 19 cigarettes per day.

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{3}=5, a_{4}=8$$

Write the indicated tem of each binomial expansion. Sixth term of \((4 h-j)^{8}\)

Find the sum for each series. $$\sum_{i=2}^{5}(6-3 i)$$

The table gives the results of a survey of \(14,000\) college students who were cigarette smokers in a recent year. $$\begin{array}{|l|c|} \hline \begin{array}{l} \text { Number of Cigarettes } \\\ \text { per Day } \end{array} & \begin{array}{c} \text { Percent } \\\ \text { (as a decimal) } \end{array} \\ \hline \text { Less than } 1 & 0.45 \\\ 1 \text { to } 9 & 0.24 \\ 10 \text { to } 19 & 0.20 \\ \text { A pack of } 20 \text { or more } & 0.11 \end{array}$$ Using the percents as probabilinies, approximate the probability that, out of 10 of these shudent smokers selected at random, the following were true. No more than 3 smoked less than 1 cigarette per day.

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{2}=9, a_{4}=13$$

Use the fundamental principle of counting or permutations to solve each problem. A concert to raise money for an economics prize is to consist of 5 works: 2 overtures, 2 sonatas, and a piano concerto. In how many ways can a program with these 5 works be arranged?

Write the indicated tem of each binomial expansion. Eighth term of \((2 c-3 d)^{14}\)

Use a formula to find the sum of each series. $$\sum_{j=2}^{7} \frac{1}{3}(4)^{j-1}$$

Find the sum for each series. $$\sum_{i=3}^{7}(5 i+2)$$

A die is rolled 12 times. Approximate the probability of rolling the following. Exactly 12 ones

Solve each problem. Tower of Hanoi A pile of \(n\) rings, each smaller than the one below it, is on a peg on a board. Two other pegs are attached to the board. In the game called the Tower of Hanoi puzzle, all the rings must be moved, one at a time, to a different peg with no ring ever placed on top of a smaller ring. Find the least number of moves that would be required. Prove your result by mathematical induction. (Figure can't copy)

Find the sum of the first 10 terms of each arithmetic sequence. $$5,9,13, \dots$$

Write the indicated tem of each binomial expansion. Fifteenth term of \(\left(a^{2}+b\right)^{22}\)