Chapter 12: Q 21. (page 830)

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

Short Answer

Expert verified

The given statement is shown.

Step by step solution

Step 1. Given information

The statement is $n^{2} - n + 2$

Step 2. Show that the given statement is true.

Put $n = 1$ in $n^{2} - n + 2$ to check if the statement is true for $n = 1$ .

width="149">(1)2-1+2=1-1+2=2

2 is divisible by 2.

Thus, the statement is true for width="37">n=1.

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

Thus, $k^{2} - k + 2$ is divisible by 2

Find whether $(k + 1)^{2} - (k + 1) + 2$ is divisible by 2.

$\begin{array}{rcl} (k + 1)^{2} - (k + 1) + 2 & = & k^{2} + 2 k + 1 - k - 1 + 2 \\ = & k^{2} - k + 2 + 2 k \\ = & (k^{2} - k + 2) + 2 k \end{array}$

Since $k^{2} - k + 2$ is divisible by 2 and $2 k$ is also divisible by $2, (k + 1)^{2} - (k + 1) + 2$ is divisible by 2 . Thus, the statement is true for width="34">k+1

Since both the conditions of the Principle of Mathematical Induction are satisfied, the statement is true for all natural numbers.

Therefore, $n^{2} - n + 2$ is divisible by 2

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. Show that the given statement is true.

One App. One Place for Learning.

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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.n2-n+2is divisible by2

Short Answer

Step by step solution

Step 1. Given information

Step 2. Show that the given statement is true.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Probability and Statistics

Mechanics Maths

Calculus

Geometry

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