BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem

In Exercises 35 48 , use mathematical induction to prove that the given statement is true for all positive integers n .

5 is a factor of 7 n 2 n .

To determine

To prove: 5 is a factor of 7n2n for all positive integers n.

Explanation

Formula used:

Mathematical Induction:

The given statement Pn is true for all positive integers n if,

a. Pn is true for n=1

b. The truth of Pk always implies that Pk+1 is true.

Proof:

Consider the statement 5 is a factor of 7n2n for all positive integers n.

By mathematical induction,

a. For n=1,

As, 7n2n=7121=5.

Since, 5 is a factor of 5.

Thus, 5 is a factor of 7n2n for n=1.

Therefore, the statement is true for n=1.

b. Assume that the statement is true for n=k.

That is, 5 is a factor of 7k2k.

7k2k=5z, for some integer z.

To show the statement is true for n=k+1.

Now,

7k+12k+1

=(7k7)(2k2)

By subtracting and adding 7k2,

=(7k

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started
Sect-2.1 P-1ESect-2.1 P-2ESect-2.1 P-3ESect-2.1 P-4ESect-2.1 P-5ESect-2.1 P-6ESect-2.1 P-7ESect-2.1 P-8ESect-2.1 P-9ESect-2.1 P-10ESect-2.1 P-11ESect-2.1 P-12ESect-2.1 P-13ESect-2.1 P-14ESect-2.1 P-15ESect-2.1 P-16ESect-2.1 P-17ESect-2.1 P-18ESect-2.1 P-19ESect-2.1 P-20ESect-2.1 P-21ESect-2.1 P-22ESect-2.1 P-23ESect-2.1 P-24ESect-2.1 P-25ESect-2.1 P-26ESect-2.1 P-27ESect-2.1 P-28ESect-2.1 P-29ESect-2.1 P-30ESect-2.1 P-31ESect-2.1 P-32ESect-2.1 P-33ESect-2.1 P-34ESect-2.1 P-35ESect-2.2 P-1ESect-2.2 P-2ESect-2.2 P-3ESect-2.2 P-4ESect-2.2 P-5ESect-2.2 P-6ESect-2.2 P-7ESect-2.2 P-8ESect-2.2 P-9ESect-2.2 P-10ESect-2.2 P-11ESect-2.2 P-12ESect-2.2 P-13ESect-2.2 P-14ESect-2.2 P-15ESect-2.2 P-16ESect-2.2 P-17ESect-2.2 P-18ESect-2.2 P-19ESect-2.2 P-20ESect-2.2 P-21ESect-2.2 P-22ESect-2.2 P-23ESect-2.2 P-24ESect-2.2 P-25ESect-2.2 P-26ESect-2.2 P-27ESect-2.2 P-28ESect-2.2 P-29ESect-2.2 P-30ESect-2.2 P-31ESect-2.2 P-32ESect-2.2 P-33ESect-2.2 P-34ESect-2.2 P-35ESect-2.2 P-36ESect-2.2 P-37ESect-2.2 P-38ESect-2.2 P-39ESect-2.2 P-40ESect-2.2 P-41ESect-2.2 P-42ESect-2.2 P-43ESect-2.2 P-44ESect-2.2 P-45ESect-2.2 P-46ESect-2.2 P-47ESect-2.2 P-48ESect-2.2 P-49ESect-2.2 P-50ESect-2.2 P-51ESect-2.2 P-52ESect-2.2 P-53ESect-2.2 P-54ESect-2.2 P-55ESect-2.2 P-56ESect-2.2 P-57ESect-2.3 P-1TFESect-2.3 P-2TFESect-2.3 P-3TFESect-2.3 P-4TFESect-2.3 P-5TFESect-2.3 P-6TFESect-2.3 P-7TFESect-2.3 P-8TFESect-2.3 P-9TFESect-2.3 P-10TFESect-2.3 P-1ESect-2.3 P-2ESect-2.3 P-3ESect-2.3 P-4ESect-2.3 P-5ESect-2.3 P-6ESect-2.3 P-7ESect-2.3 P-8ESect-2.3 P-9ESect-2.3 P-10ESect-2.3 P-11ESect-2.3 P-12ESect-2.3 P-13ESect-2.3 P-14ESect-2.3 P-15ESect-2.3 P-16ESect-2.3 P-17ESect-2.3 P-18ESect-2.3 P-19ESect-2.3 P-20ESect-2.3 P-21ESect-2.3 P-22ESect-2.3 P-23ESect-2.3 P-24ESect-2.3 P-25ESect-2.3 P-26ESect-2.3 P-27ESect-2.3 P-28ESect-2.3 P-29ESect-2.3 P-30ESect-2.3 P-31ESect-2.3 P-32ESect-2.3 P-33ESect-2.3 P-34ESect-2.3 P-35ESect-2.3 P-36ESect-2.3 P-37ESect-2.3 P-38ESect-2.3 P-39ESect-2.3 P-40ESect-2.3 P-41ESect-2.3 P-42ESect-2.3 P-43ESect-2.3 P-44ESect-2.3 P-45ESect-2.3 P-46ESect-2.3 P-47ESect-2.3 P-48ESect-2.3 P-49ESect-2.4 P-1TFESect-2.4 P-2TFESect-2.4 P-3TFESect-2.4 P-4TFESect-2.4 P-5TFESect-2.4 P-6TFESect-2.4 P-7TFESect-2.4 P-8TFESect-2.4 P-9TFESect-2.4 P-10TFESect-2.4 P-11TFESect-2.4 P-12TFESect-2.4 P-13TFESect-2.4 P-1ESect-2.4 P-2ESect-2.4 P-3ESect-2.4 P-4ESect-2.4 P-5ESect-2.4 P-6ESect-2.4 P-7ESect-2.4 P-8ESect-2.4 P-9ESect-2.4 P-10ESect-2.4 P-11ESect-2.4 P-12ESect-2.4 P-13ESect-2.4 P-14ESect-2.4 P-15ESect-2.4 P-16ESect-2.4 P-17ESect-2.4 P-18ESect-2.4 P-19ESect-2.4 P-20ESect-2.4 P-21ESect-2.4 P-22ESect-2.4 P-23ESect-2.4 P-24ESect-2.4 P-25ESect-2.4 P-26ESect-2.4 P-27ESect-2.4 P-28ESect-2.4 P-29ESect-2.4 P-30ESect-2.4 P-31ESect-2.4 P-32ESect-2.4 P-33ESect-2.4 P-34ESect-2.4 P-35ESect-2.5 P-1TFESect-2.5 P-2TFESect-2.5 P-3TFESect-2.5 P-4TFESect-2.5 P-5TFESect-2.5 P-6TFESect-2.5 P-7TFESect-2.5 P-1ESect-2.5 P-2ESect-2.5 P-3ESect-2.5 P-4ESect-2.5 P-5ESect-2.5 P-6ESect-2.5 P-7ESect-2.5 P-8ESect-2.5 P-9ESect-2.5 P-10ESect-2.5 P-11ESect-2.5 P-12ESect-2.5 P-13ESect-2.5 P-14ESect-2.5 P-15ESect-2.5 P-16ESect-2.5 P-17ESect-2.5 P-18ESect-2.5 P-19ESect-2.5 P-20ESect-2.5 P-21ESect-2.5 P-22ESect-2.5 P-23ESect-2.5 P-24ESect-2.5 P-25ESect-2.5 P-26ESect-2.5 P-27ESect-2.5 P-28ESect-2.5 P-29ESect-2.5 P-30ESect-2.5 P-31ESect-2.5 P-32ESect-2.5 P-33ESect-2.5 P-34ESect-2.5 P-35ESect-2.5 P-36ESect-2.5 P-37ESect-2.5 P-38ESect-2.5 P-39ESect-2.5 P-40ESect-2.5 P-41ESect-2.5 P-42ESect-2.5 P-43ESect-2.5 P-44ESect-2.5 P-45ESect-2.5 P-46ESect-2.5 P-47ESect-2.5 P-48ESect-2.5 P-49ESect-2.5 P-50ESect-2.5 P-51ESect-2.5 P-52ESect-2.5 P-53ESect-2.5 P-54ESect-2.5 P-55ESect-2.5 P-56ESect-2.5 P-57ESect-2.5 P-58ESect-2.6 P-1TFESect-2.6 P-2TFESect-2.6 P-3TFESect-2.6 P-4TFESect-2.6 P-5TFESect-2.6 P-6TFESect-2.6 P-7TFESect-2.6 P-8TFESect-2.6 P-1ESect-2.6 P-2ESect-2.6 P-3ESect-2.6 P-4ESect-2.6 P-5ESect-2.6 P-6ESect-2.6 P-7ESect-2.6 P-8ESect-2.6 P-9ESect-2.6 P-10ESect-2.6 P-11ESect-2.6 P-12ESect-2.6 P-13ESect-2.6 P-14ESect-2.6 P-15ESect-2.6 P-16ESect-2.6 P-17ESect-2.6 P-18ESect-2.6 P-19ESect-2.6 P-20ESect-2.6 P-21ESect-2.6 P-22ESect-2.6 P-23ESect-2.6 P-24ESect-2.6 P-25ESect-2.6 P-26ESect-2.7 P-1TFESect-2.7 P-2TFESect-2.7 P-3TFESect-2.7 P-4TFESect-2.7 P-1ESect-2.7 P-2ESect-2.7 P-3ESect-2.7 P-4ESect-2.7 P-5ESect-2.7 P-6ESect-2.7 P-7ESect-2.7 P-8ESect-2.7 P-9ESect-2.7 P-10ESect-2.7 P-11ESect-2.7 P-12ESect-2.7 P-13ESect-2.7 P-14ESect-2.7 P-15ESect-2.7 P-16ESect-2.7 P-17ESect-2.7 P-18ESect-2.7 P-19ESect-2.7 P-20ESect-2.7 P-21ESect-2.7 P-22ESect-2.7 P-23ESect-2.7 P-24ESect-2.7 P-25ESect-2.7 P-26ESect-2.8 P-1TFESect-2.8 P-2TFESect-2.8 P-3TFESect-2.8 P-1ESect-2.8 P-2ESect-2.8 P-3ESect-2.8 P-4ESect-2.8 P-5ESect-2.8 P-6ESect-2.8 P-7ESect-2.8 P-8ESect-2.8 P-9ESect-2.8 P-10ESect-2.8 P-11ESect-2.8 P-12ESect-2.8 P-13ESect-2.8 P-14ESect-2.8 P-15ESect-2.8 P-16ESect-2.8 P-17ESect-2.8 P-18ESect-2.8 P-19ESect-2.8 P-20ESect-2.8 P-21ESect-2.8 P-22ESect-2.8 P-23ESect-2.8 P-24ESect-2.8 P-25ESect-2.8 P-26E

Additional Math Solutions

Find more solutions based on key concepts

Show solutions add

Evaluate the expression sin Exercises 116. (14)2

Finite Mathematics and Applied Calculus (MindTap Course List)

State the four steps of Polyas four-step problem-solving strategy.

Mathematical Excursions (MindTap Course List)

Find the mean, median, and mode for the following scores; 8, 7, 5, 7, 0, 10, 2, 4, 11, 7, 8, 7

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Sketch the graph of the function f(x) = {4xxifx2x+2ifx2 and evaluate limxa+ f(x), limxa f(x), and limxaf(x) at ...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

If f(t) = sec t, find f"(/4).

Single Variable Calculus: Early Transcendentals, Volume I

Express the limit as a definite integral. limn1ni=1n11+(i/n)2

Single Variable Calculus: Early Transcendentals

True or False: f(x)f(a)xa may be interpreted as the instantaneous velocity of a particle at time a.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th