SOLUTION: Using math of induction show that 2^n + (-1)^n+1 is divisible by 3

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Question 1137651: Using math of induction show that 2^n + (-1)^n+1 is divisible by 3
Answer by math_helper(2461) About Me  (Show Source):
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Base case: n=1
+2%5E1+%2B+%28-1%29%5E%282%29+=+2%2B1+=+3+

Hypothesis: Assume 2%5En+%2B+%28-1%29%5E%28n%2B1%29+ is divisible by 3 for n=k

Step case: Let n=k+1:
+V+=+2%5E%28k%2B1%29+%2B+%28-1%29%5E%28%28k%2B1%29%2B1%29+
We know +W+=+2%5E%28k%29+%2B+%28-1%29%5E%28k%2B1%29+ is divisible by 3 (by hypothesis)


Look at V-W:
+V-W+=+2%5E%28k%2B1%29%2B%28-1%29%5E%28%28k%2B1%29%2B1%29+-+2%5Ek+-+%28-1%29%5E%28k%2B1%29+

= +2%5E%28k%2B1%29-2%5Ek+%2B+%28%28-1%29%5E%28%28k%2B1%29%2B1%29+-+%28-1%29%5E%28k%2B1%29%29+

The last two terms combine to -2 when k is odd, +2 when k is even, factor
out 2:
= +2%282%5E%28k%29-2%5E%28k-1%29+%2B+%28-1%29%5E%28k%29%29+

Noting that 2%5Ek-2%5E%28k-1%29+=+2%5E%28k-1%29+:
= +2+%2A+green%28+%28+2%5E%28k-1%29+%2B+%28-1%29%5Ek%29+%29+

By the hypothesis, the +green%28green%29+ factor is divisible by 3
We know W is divisible by 3 and V-W is also divisible by 3, so V = (W)+(V-W) is also divisible by 3.

DONE