Chapter 12.4, Problem 20AYU

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.

$n^3+\text{ 2}n$ is divisible by 3.

To determine

To prove: The given statement $n^3 + 2n$ is divisible by 3 is true for all natural numbers $n$ using the Principle of Mathematical Induction.

Answer to Problem 20AYU

As the statement is true for the natural number $\left(k + 1\right)$ terms, hence the statement is true for all natural numbers.

Explanation of Solution

Given:

Statements says the series $n^3 + 2n$ is divisible by 3 is true for all natural number.

Formula used:

The Principle of Mathematical Induction

Suppose that the following two conditions are satisfied with regard to a statement about natural numbers:

CONDITION I: The statement is true for the natural number 1.

CONDITION II: If the statement is true for some natural number $k$, it is also true for the next natural number $k + 1$. Then the statement is true for all natural numbers.

Proof:

Consider the statement

$n^3 + 2n$ is divisible by 3   -----(1)

Step 1: Show that statement (1) is true for the natural number $n = 1$.

That is ${\left(1\right)}^2 + 2 \left(1\right) = 3$ is divisible by 3. Hence the statement is true for the natural number $n = 1$.

Step 2: Assume that the statement is true for some natural number $k$.

That is $k^2 + 2k$ is divisible by 3   -----(1)

Step 3: Prove that the statement is true for the next natural number $k + 1$.

That is, to prove that ${\left(k + 1\right)}^3 + 2 \left(k + 1\right)$ is divisible by 3

Consider

${\left(k + 1\right)}^3 + 2 \left(k + 1\right) = k^3 + 1 + 3k^2 + 3k + 2k + 2 = \left(k^3 + 2k\right) + 3k + 3 = \left(k^3 + 2k\right) + 3 \left(k + 1\right)$

From equation (1), first term in the above equation is divisible by 3 and second term is a multiple of 3.

Hence $\left(k^3 + 2k\right) + 3 \left(k + 1\right)$ is divisible by 2.

As the statement is true for the natural number $\left(k + 1\right)$ terms, hence the statement is true for all natural numbers.