(-1)^n+(-1)^2n+(-1)^2n+1+(-1)^4n+2

2 min read Jun 16, 2024

Simplifying the Expression (-1)^n + (-1)^2n + (-1)^2n+1 + (-1)^4n+2

This article aims to simplify the expression (-1)^n + (-1)^2n + (-1)^2n+1 + (-1)^4n+2 by understanding the properties of exponents and simplifying the terms.

Understanding the Properties of Exponents

(-1)^even number = 1: Any even power of -1 results in 1.
(-1)^odd number = -1: Any odd power of -1 results in -1.

Simplifying the Expression

Let's analyze each term in the expression:

(-1)^n: This term depends on the value of 'n'. If 'n' is even, it equals 1. If 'n' is odd, it equals -1.
(-1)^2n: This term will always be 1, as 2n is always an even number.
(-1)^2n+1: This term will always be -1, as 2n+1 is always an odd number.
(-1)^4n+2: This term will always be 1, as 4n+2 is always an even number.

Combining the Terms

Now, we can combine the simplified terms:

(-1)^n + (-1)^2n + (-1)^2n+1 + (-1)^4n+2 = (-1)^n + 1 - 1 + 1 = (-1)^n + 1

Conclusion

The simplified form of the expression (-1)^n + (-1)^2n + (-1)^2n+1 + (-1)^4n+2 is (-1)^n + 1. This expression will evaluate to 0 when 'n' is odd and 2 when 'n' is even.