document.write( "Question 1137651: Using math of induction show that 2^n + (-1)^n+1 is divisible by 3 \n" ); document.write( "

Algebra.Com's Answer #755546 by math_helper(2461) $\"\"$ $\"About$

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Base case: n=1 \r
\n" ); document.write( "\n" ); document.write( " $\"+2%5E1+%2B+%28-1%29%5E%282%29+=+2%2B1+=+3+\"$
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\n" ); document.write( "\n" ); document.write( "Hypothesis: Assume $\"2%5En+%2B+%28-1%29%5E%28n%2B1%29+\"$ is divisible by 3 for n=k
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\n" ); document.write( "\n" ); document.write( "Step case: Let n=k+1:\r
\n" ); document.write( "\n" ); document.write( " $\"+V+=+2%5E%28k%2B1%29+%2B+%28-1%29%5E%28%28k%2B1%29%2B1%29+\"$ \r
\n" ); document.write( "\n" ); document.write( "We know $\"+W+=+2%5E%28k%29+%2B+%28-1%29%5E%28k%2B1%29+\"$ is divisible by 3 (by hypothesis)
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\n" ); document.write( "\n" ); document.write( "Look at V-W:
\n" ); document.write( " $\"+V-W+=+2%5E%28k%2B1%29%2B%28-1%29%5E%28%28k%2B1%29%2B1%29+-+2%5Ek+-+%28-1%29%5E%28k%2B1%29+\"$

\n" ); document.write( " = $\"+2%5E%28k%2B1%29-2%5Ek+%2B+%28%28-1%29%5E%28%28k%2B1%29%2B1%29+-+%28-1%29%5E%28k%2B1%29%29+\"$

\n" ); document.write( "The last two terms combine to -2 when k is odd, +2 when k is even, factor
\n" ); document.write( "out 2:
\n" ); document.write( " = $\"+2%282%5E%28k%29-2%5E%28k-1%29+%2B+%28-1%29%5E%28k%29%29+\"$

\n" ); document.write( "Noting that $\"2%5Ek-2%5E%28k-1%29+=+2%5E%28k-1%29+\"$ :
\n" ); document.write( " = $\"+2+%2A+green%28+%28+2%5E%28k-1%29+%2B+%28-1%29%5Ek%29+%29+\"$
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\n" ); document.write( "\n" ); document.write( "By the hypothesis, the $\"+green%28green%29+\"$ factor is divisible by 3 \r
\n" ); document.write( "\n" ); document.write( "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.
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\n" ); document.write( "\n" ); document.write( " DONE\r
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