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Prove that 7^(2n)+16n-1 is divisible by 64
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\n" ); document.write( "we prove this by using mathematical induction
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\n" ); document.write( "For n=1, we have 7^2 +16 -1 = 64 which is divisible by 64
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\n" ); document.write( "For n=k, we assume that 7^(2k)+16k-1 is divisible by 64
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\n" ); document.write( "We must check for n=k+1, is the statement true
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\n" ); document.write( "7^(2(k+1)) +16(k+1) -1 = 49 * 7^2k + 16k +15
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\n" ); document.write( "we adjust this expression by adding and subtracting terms, that is
\n" ); document.write( "we add 49*16k, -49 but we have to subtract 49*16k and -49
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\n" ); document.write( "we have the following
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\n" ); document.write( "49*7^2k +16k +15 = 49*7^2k + 49*16k -49 -49*16k -49 +16k +15 =
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\n" ); document.write( "49(7^2k +16k -1) - 49(16k-1) +16k +15 =
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\n" ); document.write( "49(7^k +16k -1) - 48*16k +64
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\n" ); document.write( "all these terms are divisible by 64, so we are done :-)
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