Exploring the Evenness of n2n for Any Positive Integer n

Introduction

In this article, we delve into the fascinating property of the expression n2n for any positive integer n. We will explore how, regardless of whether n is odd or even, the result of n2n is always an even number. Let's dive into the reasoning behind this phenomenon.

When is n2n Even?

To understand why n2n is always even, let's break it down into clear cases based on the parity of n (whether it is odd or even).

Case 1: n is Even

If n is an even number, we can express it as n 2k for some integer k. Then,

n2n (2k)2(2k 1) 4k2(2k 1) 2(2k2(2k 1))

Since the expression 2k2(2k 1) is an integer, the product 2(2k2(2k 1)) is clearly even. Thus, if n is even, n2n is even.

Case 2: n is Odd

If n is an odd number, we can express it as n 2k 1 for some integer k. Then,

n2n (2k 1)2(2k 2) 4k2 4k 1(2k 2) 4k2 8k 2 2(2k2 4k 1)

Since the expression 2k2 4k 1 is an integer, the product 2(2k2 4k 1) is also even. Thus, if n is odd, n2n is even.

Alternative Perspective

Another way to view the expression n2n is by considering its form as nn-1. This form helps us see that one of the factors is always even while the other is always odd.

For any positive integer n, the product nn-1 must be even because one of the factors is even and the other is odd. Multiplying an even number by any integer results in an even number.

Conclusion

In conclusion, the expression n2n is an even number for any positive integer n. This property holds true whether n is odd or even, and it can be verified through various algebraic manipulations and reasoning. Understanding this concept not only enriches our knowledge of number theory but also highlights the elegance and consistency of mathematical principles.

Additional Insights

Let's explore the broader implications of this phenomenon and address a related question: is this property true for all integers, not just positive ones?

Yes, the property holds for all integers, positive or negative. The proof is similar to the positive integer case:

If n is even, then n 2k, and n2n 4k2(2k 1) 2(2k2(2k 1)), which is even. If n is odd, then n 2k 1, and n2n 4k2 4k 1(2k 2) 4k2 8k 2 2(2k2 4k 1), which is even. This property underscores the fundamental principles of even and odd numbers in mathematics.