any odd integer????

Question

Prove that is x is any odd integer then ?(-1)?^x = -1

Answer 1

We can show this using induction

if x is odd then x + 2 is also odd and x - 2 is also odd

Let x = 1. We have (-1)^x = (-1)¹ = -1

Assume n is odd and (-1)ⁿ = -1. We shall show (-1)ⁿ⁺² = -1

(-1)ⁿ⁺² = (-1)ⁿ * (-1)² = -1 * (-1)² = -1 * 1 = -1

Next we shall show (-1)^n-2 = -1

(-1)^n-2 = (-1)ⁿ * (-1)^-2 = -1 * (1/(-1)²) = -1 * (1/1) = -1 * 1 = -1

 

We can also show this by algebra:

is x is odd x - 1 is even

(-1)^x = -1 * (-1)^x-1

Since x-1 is even (-1)^x-1 = 1

Then

(-1)^x = -1 * 1 = -1

Answer 2

 odd integer: 2n +1 for n =( 0.1. 2.3, .....)

 

 

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