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Prove by induction that 1 - (-2)^(n+1) is always divisible by 3?

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Prove by induction that 1 - (-2)^(n+1) is always divisible by 3?

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  1. For n = 1 we have 1 - (-2)^2 = 1 - 4 = -3, which is clearly divisible by 3.

    Assume that 1 - (-2)^(n+1) = 3k.  Now 1 - (-2)^(n+2) = 1 - (-2)(-2)^(n+1) = 1 - (-2)^(n+1) + 3(-2)^n+1) = 3[k + (-2)^(n+1)], so the result is true for n = 1.

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