Chapter 12: Q 20. (page 830)

In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.
$n^{3} + 2 n$ is divisible by $3$

Short Answer

Expert verified

The given statement is shown.

Step by step solution

Step 1. Given information

The given expression is $n^{3} + 2 n$

Step 2. Show that the given statement is correct.

First we consider the statement for $n = 1$

Because $1^{3} + 2.1 = 3$ is divisible by 3 , the statement is true for width="37">n=1

Now we consider that the statement holds for some $k$

We need to show that $(k + 1)^{3} + 2 (k + 1)$ is divisible by 3

width="305">(k+1)3+2(k+1)=k3+3k2+3k+1+2k+2=k3+2k+3k2+3k+3=k3+2k+3k2+k+1

Since $3 (k^{2} + k + 1)$ is divisible by 3 and $k^{3} + 2 k$ is divisible by 3 , it follows that $(k + 1)^{3} + 2 (k + 1)$ is divisible by 3.

As a result statement 1 is true for all natural numbers $n$

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In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.
$n^{3} + 2 n$ is divisible by $3$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Show that the given statement is correct.

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In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.n3+2n is divisible by3

Short Answer

Step by step solution

Step 1. Given information

Step 2. Show that the given statement is correct.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Geometry

Probability and Statistics

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.
$n^{3} + 2 n$ is divisible by $3$