Chapter 8

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(x-3 y)^{5}$$

Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first 11 terms of the geometric sequence: $$4,-12,36,-108, \dots$$

Involve a deck of 52 cards. A poker hand consists of five cards. a. Find the total number of possible five-card poker hands. b. A diamond flush is a five-card hand consisting of all diamonds. Find the number of possible diamond flushes. c. Find the probability of being dealt a diamond flush.

Use the Fundamental Counting Principle to solve The model of the car you are thinking of buying is available in nine different colors and three different styles (hatchback, sedan, or station wagon). In how many ways can you order the car?

In Exercises \(23-34,\) write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$a_{1}=-20, d=-4$$

In Exercises 25-34, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$\sum_{i=1}^{n} 5 \cdot 6^{i}=6\left(6^{n}-1\right)$$

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(2 a+b)^{6}$$

Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first 14 terms of the geometric sequence: $$-\frac{3}{2}, 3,-6,12, \ldots$$

Find each indicated sum. $$\sum_{i=1}^{6} 5 i$$

Involve a deck of 52 cards. If you are dealt 3 cards from a shuffled deck of 52 cards, find the probability that all 3 cards are picture cards.

Use the Fundamental Counting Principle to solve A popular brand of pen is available in three colors (red, green, or blue) and four writing tips (bold, medium, fine, or micro). How many different choices of pens do you have with this brand?

In Exercises \(23-34,\) write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$a_{1}=-70, d=-5$$

In Exercises 25-34, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$\sum_{i=1}^{n} 7 \cdot 8^{i}=8\left(8^{n}-1\right)$$

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(a+2 b)^{6}$$

Find each indicated sum. $$\sum_{i=1}^{6} 7 i$$

Use the Fundamental Counting Principle to solve An ice cream store sells two drinks (sodas or milk shakes), in four sizes (small, medium, large, or jumbo), and five flavors (vanilla, strawberry, chocolate, coffee, or pistachio). In how many ways can a customer order a drink?

In Exercises \(23-34,\) write a formula for the general term (the nth term of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$a_{n}=a_{n-1}+3, a_{1}=4$$

In Exercises 25-34, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$n+2>n$$

In Exercises 31-38, write the first three terms in each binomial expansion, expressing the result in simplified form. $$(x+2)^{8}$$

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. $$ \sum_{i=1}^{8} 3^{i} $$

Find each indicated sum. $$\sum_{i=1}^{4} 2 i^{2}$$

Use the Fundamental Counting Principle to solve A restaurant offers the following lunch menu. $$ \begin{array}{llll} \text { Main Course } & \text { Vegetables } & \text { Beverages } & \text { Desserts } \\ \text { Ham } & \text { Potatoes } & \text { Coffee } & \text { Cake } \\ \text { Chicken } & \text { Peas } & \text { Tea } & \text { Pie } \\ \text { Fish } & \text { Green beans } & \text { Milk } & \text { Ice cream } \\\ \text { Beef } & & \text { Soda } & \end{array} $$ If one item is selected from each of the four groups, in how many ways can a meal be ordered? Describe two such orders.

In Exercises \(23-34,\) write a formula for the general term (the nth term of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$a_{n}=a_{n-1}+5, a_{1}=6$$

In Exercises 25-34, use mathematical induction to prove that each statement is true for every positive integer \(n.\) If \(0 < x < 1,\) then \(0 < x^{n} < 1\).

In Exercises 31-38, write the first three terms in each binomial expansion, expressing the result in simplified form. $$(x+3)^{6}$$

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. $$\sum_{i=1}^{6} 4^{i}$$

Find each indicated sum. $$\sum_{-1}^{5} i^{3}$$

Use the Fundamental Counting Principle to solve You are taking a multiple-choice test that has five questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each question and leave nothing blank, in how many ways can you answer the questions?

In Exercises 25-34, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$(a b)^{n}=a^{n} b^{n}$$

In Exercises 31-38, write the first three terms in each binomial expansion, expressing the result in simplified form. $$(x-2 y)^{10}$$

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. $$\sum_{i=1}^{10} 5 \cdot 2^{i}$$

Use the Fundamental Counting Principle to solve You are taking a multiple-choice test that has eight questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each question and leave nothing blank, in how many ways can you answer the questions?

In Exercises 25-34, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$\left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}}$$

In Exercises 31-38, write the first three terms in each binomial expansion, expressing the result in simplified form. $$(x-2 y)^{9}$$

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. $$\sum_{i=1}^{7} 4(-3)^{i}$$

Find each indicated sum. $$\sum_{i=1}^{2}(k-3)(k+2)$$

Use the Fundamental Counting Principle to solve In the original plan for area codes in \(1945,\) the first digit could be any number from 2 through \(9,\) the second digit was either 0 or \(1,\) and the third digit could be any number except \(0 .\) With this plan, how many different area codes were possible?

Find the sum of the first 20 terms of the arithmetic sequence: \(4,10,16,22, \dots\)

Explain how to use mathematical induction to prove that a statement is true for every positive integer \(n\).

In Exercises 31-38, write the first three terms in each binomial expansion, expressing the result in simplified form. $$\left(x^{2}+1\right)^{16}$$

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. $$\sum_{i=1}^{6}\left(\frac{1}{2}\right)^{i+1}$$

Find each indicated sum. $$\sum_{i=1}^{2}\left(-\frac{1}{2}\right)^{i}$$

Use the Fundamental Counting Principle to solve How many different four-letter radio station call letters can be formed if the first letter must be \(\mathrm{W}\) or \(\mathrm{K} ?\)

Find the sum of the first 25 terms of the arithmetic sequence: \(7,19,31,43, \dots\)

In Exercises 31-38, write the first three terms in each binomial expansion, expressing the result in simplified form. $$\left(x^{2}+1\right)^{17}$$

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. $$\sum_{i=1}^{6}\left(\frac{1}{3}\right)^{i+1}$$

Find each indicated sum. $$\sum_{i=2}^{4}\left(-\frac{1}{3}\right)^{i}$$

Use the Fundamental Counting Principle to solve Six performers are to present their comedy acts on weekend evening at a comedy club. One of the performers insists on being the last stand-up comic of the evening. If this performer'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 king.

Find the sum of the first 50 terms of the arithmetic sequence: \(-10,-6,-2,2, \dots\)