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

## 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) = -1We 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}= -1Assume 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 = -1Next we shall show (-1)

^{n-2}= -1(-1)

^{n-2}= (-1)^{n}* (-1)^{-2}= -1 * (1/(-1)^{2}) = -1 * (1/1) = -1 * 1 = -1We 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}= 1Then

(-1)

^{x}= -1 * 1 = -1