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any odd integer????

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2 Answers

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