document.write( "Question 1179358: Use mathematical induction to prove each statement is true for all positive integers n:
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Algebra.Com's Answer #808892 by math_helper(2461) $\"\"$ $\"About$

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document.write( "Use mathematical induction to prove each statement is true for all positive integers n:
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document.write( "5^(n)-1 is divisible by 4
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document.write( "n^(2)-n is divisible by 2
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document.write( "n|d means \"n divides d\"\r
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document.write( "n=0:  ,  4 | 0
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document.write( "n=1:   ,  4 | 4\r
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document.write( "Assume true for n=k: i.e.  |    (hypothesis)\r
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document.write( "Let n=k+1:  
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document.write( "4 |   (by the hypothesis) so  |  as well.\r
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document.write( "Therefore,  |  (if 4|P then 4|(P+4)) 
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document.write( "and the proof is complete.\r
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document.write( "The other proof does not require induction  is always even, which is divisible by 2.   Follow the steps I did in first problem if you must have a proof by induction (show true for base case, assume hypothesis (n=k), and then show it leads to truth of the step case where n=k+1)\r
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