Chapter 12.4, Problem 21AYU

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

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

To determine

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

Answer to Problem 21AYU

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 $n^2 - n + 2$ is divisible by 2 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

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

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

That is ${\left(1\right)}^2 - 1 + 2 = 2$ is divisible by 2. 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 $k^2 - k + 2$ is divisible by 2   -----(2)

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

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

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

From equation (2), first term in the above equation is divisible by 2 and second term is a multiple of 2.

Hence ${\left(k + 1\right)}^2 - \left(k + 1\right) + 2$ 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.