Chapter 11

Use mathematical induction to prove that each statement is true for every positive integer n. \(\sum_{i=1}^{n} 7 \cdot 8^{i}=8\left(8^{n}-1\right)\)

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. $$ a_{1}=-70, d=-5 $$

Involve a deck of 52 cards. If necessary, refer to the picture of a deck of cards, Figure 11.12 on page 1130 . 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.

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

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ (x+2)^{8} $$

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

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?

Use mathematical induction to prove that each statement is true for every positive integer n. \(n+2>n\)

Involve a deck of 52 cards. If necessary, refer to the picture of a deck of cards, Figure 11.12 on page 1130 . If you are dealt 3 cards from a shuffled deck of 52 cards, find the probability that all 3 cards are picture cards.

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

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ (x+3)^{8} $$

find each indicated sum. $$ \sum_{i=1}^{5} i^{3} $$

Use mathematical induction to prove that each statement is true for every positive integer n. If \(0

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

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

find each indicated sum. $$ \sum_{k=1}^{5} k(k+4) $$

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?

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

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

find each indicated sum. $$ \sum_{k=1}^{4}(k-3)(k+2) $$

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?

Use mathematical induction to prove that each statement is true for every positive integer n. \(\left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}}\)

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

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

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

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.

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

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

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

How many different four-letter radio station call letters can be formed if the first letter must be W or K?

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

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

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

find each indicated sum. $$ \sum_{i=5}^{9} 11 $$

Six performers are to present their comedy acts on a 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?

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

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

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

find each indicated sum. $$ \sum_{i=3}^{7} 12 $$

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?

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

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

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

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

find each indicated sum. $$ \sum_{i=0}^{4} \frac{(-1)^{i}}{i !} $$

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: • Mark had told him about the foxes. • John looked out the window. • Could it be a fox? • However, nobody had seen one for months. • 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”?

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

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