Precalculus: Graphical, Numerical, Algebraic 8th Edition solutions

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 13, expand the product.n1n + 52

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 13, expand the product.1n + 221n - 32

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 13, expand the product.k1k + 121k + 22

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 46, factor the polynomial.n2 + 2n - 3

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 46, factor the polynomial.k 3 + 3k 2 + 3k + 1

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 46, factor the polynomial.n3 - 3n2 + 3n - 1

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 710, evaluate the function at the given domain values or variable expressions.1x2 = x + 4; 112, 1t2, 1t + 12

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 710, evaluate the function at the given domain values or variable expressions.1n2 = n n + 1 ; 112, 1k2, 1k + 12

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 710, evaluate the function at the given domain values or variable expressions.P1n2 = 2n 3n + 1 ; P112, P1k2, P1k + 12

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 710, evaluate the function at the given domain values or variable expressions.P1n2 = 2n2 - n - 3; P112, P1k2, P1k + 12

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 14, use mathematical induction to prove that the statement holds for all positive integers.2 + 4 + 6 + + 2n = n2 + n

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 14, use mathematical induction to prove that the statement holds for all positive integers.8 + 10 + 12 + + 12n + 62 = n2 + 7n

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 14, use mathematical induction to prove that the statement holds for all positive integers.6 + 10 + 14 + + 14n + 22 = n12n + 42

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 14, use mathematical induction to prove that the statement holds for all positive integers.14 + 18 + 22 + + 14n + 102 = 2n1n + 62

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, state an explicit rule for the nth term of the recursively defined sequence. Then use mathematical induction to prove the rule.an = an-1 + 5, a1 = 3

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, state an explicit rule for the nth term of the recursively defined sequence. Then use mathematical induction to prove the rule.an = an-1 + 2, a1 = 7

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, state an explicit rule for the nth term of the recursively defined sequence. Then use mathematical induction to prove the rule.an = 3an-1, a1 = 2

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, state an explicit rule for the nth term of the recursively defined sequence. Then use mathematical induction to prove the rule.an = 5an-1 an = 3an-1, a1 = 2 , a1 = 3

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 912, write the statements , , and (Do not write a proof.)Pn: 1 + 2 + + n = n1n + 12 2

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 912, write the statements , , and (Do not write a proof.)Pn: 12 + 32 + 52 + + 12n - 122 = n12n - 1212n + 12 3

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 912, write the statements , , and (Do not write a proof.)Pn: 1 1 # 2 + 1 2 # 3 + + 1 n # 1n + 12 = n n + 1

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 912, write the statements , , and (Do not write a proof.)Pn: a n k=1 k 4 = n1n + 1212n + 1213n2 + 3n - 12 30

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1320, use mathematical induction to prove that the statement holds for all positive integers.1 + 5 + 9 + + 14n - 32 = n12n - 12

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1320, use mathematical induction to prove that the statement holds for all positive integers.1 + 2 + 22 + + 2n-1 = 2n - 1

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1320, use mathematical induction to prove that the statement holds for all positive integers.1 1 # 2 + 1 2 # 3 + 1 3 # 4 + + 1 n1n + 12 = n n + 1

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1320, use mathematical induction to prove that the statement holds for all positive integers.1 1 # 3 + 1 3 # 5 + + 1 12n - 1212n + 12 = n 2n + 1

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1320, use mathematical induction to prove that the statement holds for all positive integers.2 n 2n

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1320, use mathematical induction to prove that the statement holds for all positive integers.3n 3n

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1320, use mathematical induction to prove that the statement holds for all positive integers.3 is a factor of n 3 + 2n

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1320, use mathematical induction to prove that the statement holds for all positive integers.6 is a factor of 7n - 1.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 21 and 22, use mathematical induction to prove that the statement holds for all positive integers. (We have already seen each proved in another way.)The sum of the first n terms of a geometric sequence with first term and common ratio r Z 1 2/11 - r2.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 21 and 22, use mathematical induction to prove that the statement holds for all positive integers. (We have already seen each proved in another way.)The sum of the first n terms of an arithmetic sequence with first term and common difference d is In Exercises 23 and 24, use mathematical

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 23 and 24, use mathematical induction to prove that the formula holds for all positive integers.Triangular Numbers n k=1 k = n1n + 12 2

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 23 and 24, use mathematical induction to prove that the formula holds for all positive integers.Sum of the First n Cubes n k=1 k 3 = n2 1n + 122 4

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2530, use the results of Exercises 2124 and Example 2 to find the sums.1 + 2 + 3 + + 500

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2530, use the results of Exercises 2124 and Example 2 to find the sums.12 + 22 + + 2502

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2530, use the results of Exercises 2124 and Example 2 to find the sums.4 + 5 + 6 + + n

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2530, use the results of Exercises 2124 and Example 2 to find the sums.13 + 23 + 33 + + 753

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2530, use the results of Exercises 2124 and Example 2 to find the sums.1 + 2 + 4 + 8 + + 234

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2530, use the results of Exercises 2124 and Example 2 to find the sums.1 + 8 + 27 + + 3375

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3134, use the results of Exercises 2124 and Example 2 to find the sum in terms of n.n k=11k 2 - 3k + 42

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3134, use the results of Exercises 2124 and Example 2 to find the sum in terms of n.a n k=1 12k 2 a + 5k - 22

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3134, use the results of Exercises 2124 and Example 2 to find the sum in terms of n.a n k=1 1k 3 - 12

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3134, use the results of Exercises 2124 and Example 2 to find the sum in terms of n.n k=1 1k 3 a + 4k - 52

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Group Activity Here is a proof by mathematical induction that any gathering of n people must all have the same blood type. (Anchor) If there is 1 person in the gathering, everyone in the gathering obviously has the same blood type. (Inductive hypothesis) Assume that any gathering of k people must all

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Writing to Learn Kitty is having trouble understanding mathematical induction proofs because she does not understand the inductive hypothesis. If we can assume it is true for k, she asks, why cant we assume it is true for n and be done with it? After all, a variable is a variable! Write a response to

Precalculus: Graphical, Numerical, Algebraic 8th Edition
True or False The goal of mathematical induction is to prove that a statement is true for all real numbers n. Justify your answer.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
True or False If Pn is the statement \((n+1)^{2}=4 n\), then P1 is true. Justify your answer.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In a proof by mathematical induction that \(1+2+3+\cdots+n=\frac{n(n+1)}{2}\) for all positive integers n, the inductive hypothesis would be to assume that (A) n=1 (B) k=1 (C) \(1=\frac{1(1+1)}{2}\) (D) \(1+2+3+\cdots+n=\frac{n(n+1)}{2}\) for all positive integers n. (E) \(1+2+3

Precalculus: Graphical, Numerical, Algebraic 8th Edition
The first step in a proof by mathematical induction is to prove (A) the anchor. (B) the inductive hypothesis. (C) the inductive step. (D) the inductive principle. (E) the inductive foundation.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Which of the following could be used to prove that \(1+3+5+\cdots+(2 n-1)=n^{2}\) for all positive integers n? I. Mathematical induction II. The formula for the sum of a finite arithmetic sequence III. The formula for the sum of a finite geometric sequence (A) I only (B) I an

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Mathematical induction can be used to prove that, for any positive integer n, \(\sum_{k=1}^{n} k^{3}=\) (A) \(\frac{n(n+1)}{2}\) (B) \(\frac{n^{2}(n+1)^{2}}{2}\) (C) \(\frac{n^{2}(n+1)^{2}}{4}\) (D) \(\frac{n^{3}(n+1)^{3}}{2}\) (E) \(\frac{n^{3}(n+1)^{3}}{8}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Use mathematical induction to prove that 2 is a factor of (n+1)(n+2) for all positive integers n.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Use mathematical induction to prove that 6 is a factor of n(n+1)(n+2) for all positive integers n. (You may assume the assertion in Exercise 43 to be true.)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Give an alternate proof of the assertion in Exercise 43 based on the fact that (n+1)(n+2) is a product of two consecutive integers.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Give an alternate proof of the assertion in Exercise 44 based on the fact that n(n+1)(n+2) is a product of three consecutive integers.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 47 and 48, use mathematical induction to prove that the statement holds for all positive integers. Fibonacci Sequence and Series \(F_{n+2}-1=\sum_{k=1}^{n} F_{k}, \text { where }\left\{F_{n}\right\}\) is the Fibonacci sequence.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 47 and 48, use mathematical induction to prove that the statement holds for all positive integers. If \(\left\{a_{n}\right\}\) is the sequence \(\sqrt{2}, \sqrt{2+\sqrt{2}}\), \(\sqrt{2+\sqrt{2+\sqrt{2}}}, \ldots, \text { then } a_{n}<2\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Let a be any integer greater than 1. Use mathematical induction to prove that \(a-1 \text { divides } a^{n}-1\) evenly for all positive integers n.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Give an alternate proof of the assertion in Exercise 49 based on the Factor Theorem of Section 2.4.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
It is not necessary to anchor a mathematical induction proof at ; we might only be interested in the integers greater than or equal to some integer c. In this case, we simply modify the anchor and inductive step as follows: Anchor: Pc is true. Inductive Step: If Pk is true for some \(k \geq

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Proving the interior angle formula Use extended mathematical induction to prove \(P_{n} \text { for } n \geq 3\). Pn: The sum of the interior angles of an n-sided polygon is \(180^{\circ}(n-2)\)