Chapter 11

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

Find the term indicated in each expansion. $$ (x+2 y)^{6} ; \text { third term } $$

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?

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

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

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

Find the term indicated in each expansion. $$(x-1)^{9} ; fifth\quad term$$

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

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?

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 \(k \geq m .\) 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.

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

Find the term indicated in each expansion. $$ (x-1)^{10} ; \text { fifth term } $$

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

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?

Find the sum of the first 80 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 \(k \geq m .\) 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.

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

Find the term indicated in each expansion. $$ \left(x^{2}+y^{3}\right)^{8} ; \text { sixth term } $$

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

Find the sum of the even integers between 21 and 45

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. \(S_{n}: \frac{1}{4}+\frac{1}{12}+\frac{1}{24}+\cdots+\frac{1}{2 n(n+1)}=?\)

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?

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

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

Find the term indicated in each expansion. $$ \left(x^{3}+y^{2}\right)^{8} ; \text { sixth term } $$

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

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?

Find the sum of the odd integers between 30 and 54

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. \(S_{n}:\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right) \cdots\left(1-\frac{1}{n+1}\right)=?\)

Express each repeating decimal as a fraction in lowest terms. $$ 0 . \overline{5}=\frac{5}{10}+\frac{5}{100}+\frac{5}{1000}+\frac{5}{10,000}+\cdots $$

Find the term indicated in each expansion. $$ \left(x-\frac{1}{2}\right)^{9} ; \text { fourth term } $$

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ 2+2^{2}+2^{3}+\cdots+2^{11} $$

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?

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. $$ \sum_{i=1}^{17}(5 i+3) $$

Fermat’s most notorious theorem, described in the section opener on page 1098, 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.

Express each repeating decimal as a fraction in lowest terms. $$ 0 . \overline{1}=\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\frac{1}{10,000}+\cdots $$

Find the term indicated in each expansion. $$ \left(x+\frac{1}{2}\right)^{8} ; \text { fourth term } $$

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ 5+5^{2}+5^{3}+\cdots+5^{12} $$

In a production of West Side Story, eight actors are considered for the male roles of Tony, Riff, and Bernardo. In how many ways can the director cast the male roles?

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. $$ \sum_{i=1}^{20}(6 i-4) $$

Determine whether the values in each table belong to an exponential function, a logarithmic function, a linear function, or a quadratic function. a) \(\begin{array}{cc}{x} & {y} \\ {0} & {7} \\ {1} & {4} \\ {2} & {1} \\\ {3} & {-2} \\ {4} & {-5}\end{array}\) b) \(\begin{array}{cc}{x} & {y} \\ {0} & {1} \\ {1} & {4} \\ {2} & {16} \\\ {3} & {64} \\ {4} & {256}\end{array}\)

Use this information to solve Exercises \(47-48 .\) 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.

Express each repeating decimal as a fraction in lowest terms. $$ 0 . \overline{47}=\frac{47}{100}+\frac{47}{10,000}+\frac{47}{1,000,000}+\cdots $$

Find the term indicated in each expansion. $$\left(x^{2}+y\right)^{22} ; the\quad term\quad containing\quad y^{14}$$

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ 1+2+3+\dots+30 $$

Nine bands have volunteered to perform at a benefit concert, but there is only enough time for five of the bands to play. How many lineups are possible?

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. $$ \sum_{i=1}^{30}(-3 i+5) $$