document.write( "Question 899100: Prove by mathematical induction that 3^(2n)-8n-1, n is a positive integer, is a multiple of 64 \n" ); document.write( "
Algebra.Com's Answer #545243 by Edwin McCravy(20056) $\"\"$ $\"About$
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Prove by mathematical induction that 3^(2n)-8n-1, n is a positive integer, is a multiple of 64
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document.write( "Let's rewrite  as  or \r\n" );
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document.write( "So we have to prove that \r\n" );
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document.write( " is a multiple of 64.\r\n" );
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document.write( "Prove true for n=1\r\n" );
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document.write( " =  = \r\n" );
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document.write( "0 is a multiple of every number so it's a multiple of 64.\r\n" );
document.write( "But since that doesn't satisfy some people, we'll prove\r\n" );
document.write( "it true for n=2 as well.\r\n" );
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document.write( " =  = \r\n" );
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document.write( "So let k be any value, such as 1 or 2, that we've proved it for.\r\n" );
document.write( "Then if that is the case then there is aome positive integer M,\r\n" );
document.write( "such that\r\n" );
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document.write( "(1)   = \r\n" );
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document.write( "Now let's investigate to see what we must end up with if we are to \r\n" );
document.write( "prove the proposition.  Let's substitute k+1 for n and see what we are\r\n" );
document.write( "going to have to end up with:\r\n" );
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document.write( "      \r\n" );
document.write( "     \r\n" );
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document.write( "(2)   \r\n" );
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document.write( "Now let's divide (1) into (2) by long division to see what we'll\r\n" );
document.write( "have to multiply the left side of (1) by to get (2).\r\n" );
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document.write( "We start with this long division, which is a little weird because\r\n" );
document.write( "we have something to the k power instead of k to some power, but \r\n" );
document.write( "that's OK:\r\n" );
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document.write( "       ----------------- \r\n" );
document.write( "9^k-8k-1)9*9^k - 8k -  9\r\n" );
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document.write( "9^k goes into 9*9^k 9 times, so we put 9 as a quotient and then \r\n" );
document.write( "multiply and subtract to find the remainder of 64k\r\n" );
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document.write( "           9 \r\n" );
document.write( "       ----------------- \r\n" );
document.write( "9^k-8k-1)9*9^k -  8k -  9\r\n" );
document.write( "        9*9^k - 72k -  9 \r\n" );
document.write( "         ----------------\r\n" );
document.write( "               64k   \r\n" );
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document.write( "So the division gives \r\n" );
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document.write( "(3)  \r\n" );
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document.write( "So that's what we have to multiply both sides of (1) by in order\r\n" );
document.write( "to get the left side of (2).  But notice that we are assuming that\r\n" );
document.write( "we already have a value of 64M for which (1) is true, so we can\r\n" );
document.write( "replace the denominator of (3) by 64M and get\r\n" );
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document.write( "(4)   = \r\n" );
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document.write( "So we multiply the left side of (1) by (3) and the right side of (1)\r\n" );
document.write( "by (4) since they are equal.\r\n" );
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document.write( "     = \r\n" );
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document.write( "And we have already shown by the long division that the left side will\r\n" );
document.write( "become the left side of (2), so we don't need to multiply it out, (but\r\n" );
document.write( "you can if you like) and we will multiply out the right side\r\n" );
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document.write( "     = \r\n" );
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document.write( "and we can show that the right side is a multiple of 64 because\r\n" );
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document.write( "                = \r\n" );
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document.write( "So since we know that the proposition holds when n=k=1 and n=k=2, then\r\n" );
document.write( "it must be true for n=k=3.  Then if it is true for n=k=3 then it is true for\r\n" );
document.write( "n=k=4, etc. etc. So it is true for all integer values of n.\r\n" );
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document.write( "Edwin
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