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

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

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

^{2n }. ( -1) = (+1) ( -1) = -1-
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Prove that is x is any odd integer then ?(-1)?^x = -1

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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)^{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

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