document.write( "Question 1183659: Hello this is a mathematical induction question i need help with.\r
\n" ); document.write( "\n" ); document.write( "show that 5^(k+2) - 7 * 5^k + 12 is divisible by 6\r
\n" ); document.write( "\n" ); document.write( "thank you \n" ); document.write( "
Algebra.Com's Answer #814071 by robertb(5830)\"\" \"About 
You can put this solution on YOUR website!
To show by induction: \"5%5E%28k%2B2%29+-+7%2A5%5Ek+%2B+12\" is divisible by 6.\r
\n" ); document.write( "\n" ); document.write( "For n = 1: \"5%5E%281%2B2%29+-+7%2A5%5E1+%2B+12+=+102+=+6%2A17\", and the statement is true for n=1.\r
\n" ); document.write( "\n" ); document.write( "Inductive hypothesis: Assume true for some n = k, i.e., let \"5%5E%28k%2B2%29+-+7%2A5%5Ek+%2B+12\" be true, and show that \r
\n" ); document.write( "\n" ); document.write( " \"5%5E%28k%2B3%29+-+7%2A5%5E%28k%2B1%29+%2B+12\" is also true.\r
\n" ); document.write( "\n" ); document.write( "Now .\r
\n" ); document.write( "\n" ); document.write( "Since 5^(k+2) - 7*5^k + 12 is divisible by 6 by the inductive hypothesis, and so is 48, it follows that
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\n" ); document.write( "\"5%5E%28k%2B3%29+-+7%2A5%5E%28k%2B1%29+%2B+12\"\r
\n" ); document.write( "\n" ); document.write( "is also divisible by 6, and the statement is proved.\r
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\n" ); document.write( "\n" ); document.write( "Addendum: Sans induction, the statement can be proven quite easily.\r
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\n" ); document.write( "\n" ); document.write( "and thus follows directly the conclusion.\r
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