Chapter 12.4, Problem 28AYU

To determine

To prove: The given statement $n^2 - n + 41$ is a prime number is true for $n = 1$ but not true for $n = 41$ using the Principle of Mathematical Induction.

Answer to Problem 28AYU

It can be seen that square number cannot be prime. Therefore the statement $n^2 - n + 41$ is a prime number is not true for $n = 41$.

Explanation of Solution

Given:

Statements says the series $n^2 - n + 41$ is a prime number is true for $n = 1$ but not true for $n = 41$.

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

$n^2 - n + 41$ is a prime number is true for $n = 1$ but not true for $n = 41$    -----(1)

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

That is ${\left(1\right)}^2 - 1 + 41 = 41$. The number 41 is a prime number. Hence the statement is true for the natural number $n = 1$.

Step 2: Let’s prove that the statement is not true for $n = 41$.

That is ${\left(41\right)}^2 - 41 + 41 = {41}^2$    -----(2)

From the above equation, it can be seen that square number cannot be prime. Therefore the statement $n^2 - n + 41$ is a prime number is not true for $n = 41$.