Chapter 8

In Exercises 31-38, write the first three terms in each binomial expansion, expressing the result in simplified form. $$\left(y^{3}-1\right)^{20}$$

Find the sum of each infinite geometric series. $$1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots$$

Find each indicated sum. $$\sum_{i=1}^{9} 11$$

Use the Fundamental Counting Principle to solve Five singers are to perform at a night club. One of the singers insists on being the last performer of the evening. If this singer's request is granted, how many different ways are there to schedule the appearances?

You are dealt one card from a 52 -card deck. Find the probability that you are not dealt a picture card.

Find the sum of the first 50 terms of the arithmetic sequence: \(-15,-9,-3,3,\)

In Exercises 31-38, write the first three terms in each binomial expansion, expressing the result in simplified form. $$\left(y^{3}-1\right)^{21}$$

Find the sum of each infinite geometric series. $$1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\dots$$

Find each indicated sum. $$\sum_{i=1}^{7} 12$$

Use the Fundamental Counting Principle to solve In the Cambridge Encyclopedia of Language (Cambridge University Press, 1987 ), author David Crystal presents five sentences that make a reasonable paragraph regardless of their order. The sentences are as follows: \(\cdot\) Mark had told him about the foxes. \(\cdot\) John looked out the window. \(\cdot\) Could it be a fox? . However, nobody had seen one for months. \(\cdot\) He thought he saw a shape in the bushes. How many different five-sentence paragraphs can be formed if the paragraph begins with "He thought he saw a shape in the bushes" and ends with "John looked out of the window"?

You are dealt one card from a 52 -card deck. Find the probability that you are dealt a 2 or a \(3 .\)

Find \(1+2+3+4+\dots+100,\) the sum of the first 100 natural numbers.

In Exercises 39-48, find the term indicated in each expansion. $$(2 x+y)^{6} ; \text { third term }$$

Find the sum of each infinite geometric series. $$3+\frac{3}{4}+\frac{3}{4^{2}}+\frac{3}{4^{3}}+\cdots$$

Find each indicated sum. $$\sum_{i=0}^{1} \frac{(-1)^{i}}{i !}$$

Use the Fundamental Counting Principle to solve A television programmer is arranging the order that five movies will be seen between the hours of 6 p.M. and 4 . A.M. Two of the movies have a G rating and they are to be shown in the first two time blocks. One of the movies is rated NC-17 and it is to be shown in the last of the time blocks, from 2 A.M. until 4 A.M. Given these restrictions, in how many ways can the five movies be arranged during the indicated time blocks?

You are dealt one card from a 52 -card deck. Find the probability that you are dealt a red 7 or a black 8 .

Find \(2+4+6+8+\dots+200,\) the sum of the first 100 positive even integers.

In Exercises 39-48, find the term indicated in each expansion. $$(x+2 y)^{6} ; \text { third term }$$

Find the sum of each infinite geometric series. $$5+\frac{5}{6}+\frac{5}{6^{2}}+\frac{5}{6^{3}}+\cdots$$

Use the formula for \(_{n} P_{r}\) to solve Exercises \(41-48\) A club with ten members is to choose three officers president, vice president, and secretary-treasurer. If each office is to be held by one person and no person can hold more than one office, in how many ways can those offices be filled?

You are dealt one card from a 52 -card deck. Find the probability that you are dealt a 7 or a red card.

Find the sum of the first 60 positive even integers.

Some statements are false for the first few positive integers, but true for some positive integer \(m\) on. In these instances, you can prove \(S_{n}\) for \(n \geq m\) by showing that \(S_{m}\) is true and that \(S_{k}\) implies \(S_{k+1}\) when \(k \geq m .\) Use this extended principle of mathematical induction to prove that each statement in Exercises \(41-42\) is true. Prove that \(n^{2}>2 n+1\) for \(n \geq 3\). Show that the formula is true for \(n=3\) and then use step 2 of mathematical induction.

In Exercises 39-48, find the term indicated in each expansion. $$(x-1)^{9} ;$$ fifth term

Find the sum of each infinite geometric series. $$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots$$

Find each indicated sum. $$\sum_{i=1}^{5} \frac{i !}{(i-1) !}$$

Use the formula for \(_{n} P_{r}\) to solve Exercises \(41-48\) A corporation has ten members on its board of directors. In how many different ways can it elect a president, vice president, secretary, and treasurer?

You are dealt one card from a 52 -card deck. Find the probability that you are dealt a 5 or a black card.

Some statements are false for the first few positive integers, but true for some positive integer \(m\) on. In these instances, you can prove \(S_{n}\) for \(n \geq m\) by showing that \(S_{m}\) is true and that \(S_{k}\) implies \(S_{k+1}\) when \(k \geq m .\) Use this extended principle of mathematical induction to prove that each statement in Exercises \(41-42\) is true. Prove that \(2^{n}>n^{2}\) for \(n \geq 5\). Show that the formula is true for \(n=5\) and then use step 2 of mathematical induction.

Find the sum of the first 80 positive even integers.

In Exercises 39-48, find the term indicated in each expansion. $$(x-1)^{10} ; \text { fifth term }$$

Find the sum of each infinite geometric series. $$3-1+\frac{1}{3}-\frac{1}{9}+\cdots$$

Find each indicated sum. $$\sum_{-1}^{5} \frac{(i+2) !}{i !}$$

Use the formula for \(_{n} P_{r}\) to solve Exercises \(41-48\) For a segment of a radio show, a disc jockey can play 7 songs. If there are 13 songs to select from, in how many ways can the program for this segment be arranged?

In Exercises \(43-44\), find \(S_{1}\) through \(S_{5}\) and then use the pattern to make a conjecture about \(S_{n}\). Prove the conjectured formula for \(S_{n}\) by mathematical induction. $$S_{n}: \frac{1}{4}+\frac{1}{12}+\frac{1}{24}+\cdots+\frac{1}{2 n(n+1)}=?$$

Find the sum of the even integers between 21 and 45

In Exercises 39-48, find the term indicated in each expansion. $$\left(x^{2}+y^{3}\right)^{8} ; \text { sixth term }$$

Find the sum of each infinite geometric series. $$\sum_{i=1}^{\infty} 8(-0.3)^{i-1}$$

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$1^{2}+2^{2}+3^{2}+\dots+15^{2}$$

Use the formula for \(_{n} P_{r}\) to solve Exercises \(41-48\) Suppose you are asked to list, in order of preference, the three best movies you have seen this year. If you saw 20 movies during the year, in how many ways can the three best be chosen and ranked?

In Exercises \(43-44\), find \(S_{1}\) through \(S_{5}\) and then use the pattern to make a conjecture about \(S_{n}\). Prove the conjectured formula for \(S_{n}\) by mathematical induction. $$S_{n}:\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right) \cdots\left(1-\frac{1}{n+1}\right)=?$$

Find the sum of the odd integers between 30 and 54

In Exercises 39-48, find the term indicated in each expansion. $$\left(x^{3}+y^{2}\right)^{8} ; \text { sixth term }$$

Find the sum of each infinite geometric series. $$\sum_{i=1}^{\infty} 12(-0.7)^{i-1}$$

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$1^{4}+2^{4}+3^{4}+\cdots+12^{4}$$

Use the formula for \(_{n} P_{r}\) to solve Exercises \(41-48\) In a race in which six automobiles are entered and there are no ties, in how many ways can the first three finishers come in?

The mathematics department of a college has 8 male professors, 11 female professors, 14 male teaching assistants, and 7 female teaching assistants. If a person is selected at random from the group, find the probability that the selected person is a professor or a male.

Fermat's most notorious theorem, described in the section opener on page \(744,\) baffled the greatest minds for more than three centuries. In \(1994,\) after ten years of work, Princeton University's Andrew Wiles proved Fermat's Last Theorem. People magazine put him on its list of "the 25 most intriguing people of the year," the Gap asked him to model jeans, and Barbara Walters chased him for an interview." Who's Barbara Walters?" asked the bookish Wiles, who had somehow gone through life without a television. Using the 1993 PBS documentary "Solving Fermat: Andrew Wiles" or information about Andrew Wiles on the Internet, research and present a group seminar on what Wiles did to prove Fermat's Last Theorem, problems along the way, and the role of mathematical induction in the proof.

For Exercises \(45-50,\) write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$\sum_{i=1}^{17}(5 i+3)$$