document.write( "Question 1183367: prove that n^2 ≤ n! for all n ≥ 4 using mathematical induction \n" ); document.write( "

Algebra.Com's Answer #813648 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "prove that n^2 ≤ n! for all n ≥ 4 using mathematical induction
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\n" ); document.write( "\n" ); document.write( "Proof (from https://people.cs.umass.edu/~barring/cs240/induction_sol.pdf )\r
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document.write( "Note first that:\r\n" );
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document.write( "  if n = 0, then 02 = 0 and 0! = 1.\r\n" );
document.write( "  if n = 1, then 12 = 1 and 1! = 1.\r\n" );
document.write( "  if n = 2, then 22 = 4 and 2! = 2.\r\n" );
document.write( "  if n = 3, then 32 = 9 and 3! = 6.\r\n" );
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document.write( "We prove by induction on n that   ≤ n!  for all  n ≥ 4.\r\n" );
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document.write( "Basis step : \r\n" );
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document.write( "     = 16 and 4! = 24\r\n" );
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document.write( "Inductive hypothesis : \r\n" );
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document.write( "    Assume for some integer k ≥ 4 that   ≤ k!\r\n" );
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document.write( "Inductive step :\r\n" );
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document.write( "       (k + 1)! = (k + 1)k! \r\n" );
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document.write( "     ≥ \r\n" );
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document.write( "     = \r\n" );
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document.write( "     ≥ \r\n" );
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document.write( "     = \r\n" );
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document.write( "     ≥ \r\n" );
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document.write( "     ≥ \r\n" );
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document.write( "     = .\r\n" );
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document.write( "According to the method of Mathematical induction, the proof is completed.\r\n" );
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