Chapter 8

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

Use a calculator to evaluate each expression. $$_{32} C_{4}$$

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

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

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. No girls

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

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}

Concept Check 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

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

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

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. No boys

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

Explain the difference between a permutation and a combination.

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

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

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. At least 3 boys

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

Use the fundamental principle of counting or permutations to solve each problem. Home Plan Choices 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?

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

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 25 and 26 to the nearest hundredth. $$\sum_{i=1}^{6} 5(2)^{i-1}$$

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

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 25 and 26 to the nearest hundredth. $$\sum_{j=2}^{7} \frac{1}{3}(4)^{j-1}$$

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. No more than 4 girls

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

Use the fundamental principle of counting or permutations to solve each problem. Auto Varieties \(\quad\) 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?

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

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

Use the formula for \(S_{n}\) to find the sum of the first five terms for each geometric sequence. Round the answers for Exercises 25 and 26 to the nearest hundredth. $$\sum_{k=1}^{4}-2\left(\frac{1}{2}\right)^{k}$$

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

Use the fundamental principle of counting or permutations to solve each problem. Radio Station Call Letters 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?

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

Find the sum of the first 10 terms of each arithmetic sequence. $$8,6,4, \dots$$

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

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

Use the fundamental principle of counting or permutations to solve each problem. Meal Choices A menu offers a choice of 3 salads, 8 main dishes, and 5 desserts. How many different 3-course meals (salad, main dish, dessert) are possible?

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

Under what conditions does the sum of the terms of an infinite geometric sequence exist?

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

Use the fundamental principle of counting or permutations to solve each problem. Names for a Baby \(\quad\) A couple having a baby has narrowed down the choice of a name for the new baby to 3 first names and 5 middle names. How many different first- and middle-name pairings are possible?

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

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{1}=10, a_{10}=5.5$$

The table gives the results of a survey of \(14,000\) college students who were cigarette smoker in a recent year. \begin{array}{l|c}\hline \text { Number of Cigarettes per Day } & \text { Percent (as a Decimal) } \\\\\hline \text { Less than 1 } & 0.45 \\\1 \text { to } 9 & 0.24 \\\10 \text { to } 19 & 0.20 \\\\\text { A pack of 20 or more } & 0.11\end{array} Using the percents as probabilities, approximate the probability that, out of 10 of these student 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. $$a_{1}=-8, a_{10}=-1.25$$

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

The number \(0.999 \ldots\) can be written as the sum of the terms of an infinite geometric sequence: \(0.9+0.09+0.009+\cdots\) Here we have \(a_{1}=0.9\) and \(r=0.1 .\) Use the formula for \(S_{\infty}\) to find this sum.

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

Use the fundamental principle of counting or permutations to solve each problem. Concert Program Arrangement \(\mathrm{A}\) 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?

Find the sum for each series. $$\sum_{i=-1}^{2} 5(2)^{i}$$

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{20}=1090, a_{20}=102$$

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