Chapter 8

The first five terms of an arithmetic sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight rems of the sequence for the graphical and numerical representations. $$ 1,3,5,7,9 $$

Evaluate the expression. \(10 !\)

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{1}=4, a_{20}=190.2 $$

Use the binomial theorem to expand each expression. $$ (2 r+3 t)^{4} $$

Find the probability of the compound event. Drawing a pair (two cards with the same value) from a standard deck of 52 cards without replacement

The series of sketches starts with an equilateral triangle having sides of length \(1 .\) In the following steps, equilateral triangles are constructed on each side of the preceding figure. The length of the sides of each new triangle is \(\frac{1}{3}\) the length of the sides of the preceding triangles. Develop a formula for the number of sides of the \(n\) th figure. Use mathematical induction to prove your answer. (SHAPE NOT COPY)

The first five terms of an arithmetic sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight rems of the sequence for the graphical and numerical representations. $$ 4,1,-2,-5,-8 $$

Evaluate the expression. \(7 !\)

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{1}=-4, a_{20}=15 $$

Use Pascal's triangle to help expand the expression. $$ (x+y)^{2} $$

Find the probability of the compound event. (Refer to Example 7.) Calculate the probability of drawing 3 hearts and 2 diamonds in a 5 -card poker hand. Assume that drawn cards are not replaced and that the 5 cards are drawn only once.

The first five terms of an arithmetic sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight rems of the sequence for the graphical and numerical representations. $$ 7.5,6,4.5,3,1.5 $$

Evaluate the expression. \(P(5,3)\)

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{1}=-2, a_{11}=50 $$

Use Pascal's triangle to help expand the expression. $$ (m+n)^{3} $$

The first five terms of an arithmetic sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight rems of the sequence for the graphical and numerical representations. $$5.1,5.5,5.9,6.3,6.7$$

Evaluate the expression. \(P(10,2)\)

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{1}=6, a_{5}=-30 $$

Use Pascal's triangle to help expand the expression. $$ (3 x+1)^{4} $$

Find the probability of the compound event. A quality-control experiment involves selecting one string of decorative lights from a box of \(20 .\) If the string is defective, the entire box of 20 is rejected. Suppose the box comtains four defective strings of lights. What is the probability of rejecting the box?

A pile of \(n\) rings, each ring smaller than the one below it, is on a peg. Two other pegs are attached to the same board as this peg. In a game called the Tower of Hanoi puzzle, all the rings must be moved to a different peg, with only one ring moved at a time and with no ring ever placed on top of a smaller ring. Find the least number of moves required. Prove your result with mathematical induction. (PICTURE NOT COPY)

The first five terms of an arithmetic sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight rems of the sequence for the graphical and numerical representations. $$5.1,5.5,5.9,6.3,6.7$$

Evaluate the expression. \(P(8,1)\)

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{2}=6, a_{12}=31 $$

Use Pascal's triangle to help expand the expression. $$ (2 x-1)^{4} $$

Explain the principle of mathematical induction.

The first five terms of an arithmetic sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight rems of the sequence for the graphical and numerical representations. $$2,4,6,8,10$$

Evaluate the expression. \(P(6,6)\)

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{8}=4, a_{10}=14 $$

Use Pascal's triangle to help expand the expression. $$ (2-x)^{5} $$

Find the probability of the compound event. A group of students is preparing for college entrance exams. It is estimated that \(50 \%\) need help with mathematics, \(45 \%\) with English, and \(25 \%\) with both. A. Draw a Venn diagram representing these data. B. Use this diagram to find the probability that a student needs help with mathematics, English, or both. C. Solve part (b) symbolically by applying a probability formula.

Explain how the generalized principle of mathematical induction differs from the principle of mathematical induction.

The first five terms of a geometric sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight terms of the sequence for the graphical and numerical representations. $$8,4,2,1, \frac{1}{2}$$

Evaluate the expression. \(P(7,3)\)

Use a formula to find the sum of the finite geometric series. $$ 1+2+4+8+16+32+64+128 $$

Use Pascal's triangle to help expand the expression. $$ (2 a+3 b)^{3} $$

In a college of 5500 students, 950 students are enrolled in English classes, 1220 in business classes, and 350 in both. If a student is chosen at random, find the probability that he or she is enrolled in an English class, a business class, or both.

When using mathematical induction, why is it important to prove that the statement holds for \(n=1 ?\)

The first five terms of a geometric sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight terms of the sequence for the graphical and numerical representations. $$32,-8,2,-\frac{1}{2}, \frac{1}{8}$$

Evaluate the expression. \(P(12,3)\)

Use a formula to find the sum of the finite geometric series. $$ 2+\frac{1}{2}+\frac{1}{8}+\frac{1}{32}+\frac{1}{128}+\frac{1}{512} $$

Use Pascal's triangle to help expand the expression. $$ \left(x^{2}+2\right)^{4} $$

In \(2002,\) a total of \(119,923\) new books and editions were published. The table lists the numbers of books published in specific areas. If a new book or edition is selected at random, find the probability that its subject area is one of those specified. $$ \begin{array}{|ll|} \hline \text { Art } & 4481 \\ \hline \text { Business } & 4539 \\ \hline \text { History } & 6818 \\ \hline \text { Music } & 1614 \\ \hline \text { Rellglon } & 6659 \\ \hline \text { Science } & 7032 \\ \hline \end{array} $$ A. Art or music B. Neither science nor religion

The first five terms of a geometric sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight terms of the sequence for the graphical and numerical representations. $$\frac{3}{4}, \frac{3}{2}, 3,6,12$$

Evaluate the expression. \(P(25,2)\)

Use a formula to find the sum of the finite geometric series. $$ 0.5+1.5+4.5+13.5+40.5+121.5+364.5 $$

Use Pascal's triangle to help expand the expression. $$ \left(5-x^{2}\right)^{3} $$

\(\quad\) In \(2004,\) the U.S. death rate per \(100,000\) people was \(817 .\) What is the probability that a person selected at random died during \(2004 ?\) (Source: Depertment of Health and Human Services.)

The first five terms of a geometric sequence are given. Find (a) numerical, (b) graphical, and (c) symbolic representations of the sequence. Include at least eight terms of the sequence for the graphical and numerical representations. $$\frac{1}{27}, \frac{1}{9}, \frac{1}{3}, 1,3$$

Evaluate the expression. \(P(20,1)\)