document.write( "Question 1183221: Hello, this is a mathematical induction question i had a hard time to prove\r
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\n" ); document.write( "a 1^2 + 3^2 + 5^2 + … + (2n − 1)^2 = (n(4n^2 - 1))/3 \n" ); document.write( "
Algebra.Com's Answer #813429 by math_helper(2461)\"\" \"About 
You can put this solution on YOUR website!

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document.write( "LHS is  \"1%5E2%2B3%5E2\"+...+\"+%282n-1%29%5E2+\"      (1)
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document.write( "RHS is  \"n%284n%5E2-1%29%2F3\"                   (2)\r
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document.write( "Base case:
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document.write( "n=1:   LHS is \"+%282%281%29-1%29%5E2+\" = 1
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document.write( "       RHS is \"+1%2A%284%2A1%5E2-1%29%2F3+\" = (4-1)/3 }}} = 1 \r
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document.write( "Base case holds.\r
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document.write( "Hypothesis:
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document.write( "       Assume LHS = RHS for  n=k             (*)\r
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document.write( "Step case: 
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document.write( "        Let n=k+1  (recall the index k counts by 1 and the 2k-1 in the LHS & RHS is what makes sure you have odd numbers only)\r
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document.write( "What you need to do now, is show LHS=RHS for n=k+1, then the proof is complete.\r
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document.write( "LHS is  \"+green%28+1%5E2%2B3%5E2+%29+\"+...+\"+green%28%282k-1%29%5E2%29+\" + \"+%282%28k%2B1%29-1%29%5E2+\" 
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document.write( "Where I have separated the (k+1)th term.  The terms in green are the n=k case, which by the hypothesis (*), can be replaced by \"k%284k%5E2-1%29%2F3\", giving:\r
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document.write( "LHS = \"+k%284k%5E2-1%29%2F3+\" + \"+%282%28k%2B1%29-1%29%5E2+\"\r
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document.write( "    ...expand and simplify... 
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document.write( "    = \"+%284k%5E3%2B12k%5E2%2B11k%2B3%29%2F3+\"\r
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document.write( "    ... factor (I used WolframAlpha, you could also guess k+1 as likely
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document.write( "factor and do the division)...\r
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document.write( "    = \"+%282k%2B1%29%282k%2B3%29%28k%2B1%29+%2F+3+\"
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document.write( "  
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document.write( "Is this last expression the same as (2)?
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document.write( "  Let u=k+1,  --> k=u-1
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document.write( "  Then the last expression above, in terms of u, is:
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document.write( "     = \"+%282%28u-1%29%2B1%29%282%28u-1%29%2B3%29u%29%2F3+\"
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document.write( "     = \"+%28%282u-1%29%282u%2B1%29u%29%2F3+\"
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document.write( "     = \"++%28%284u%5E2-1%29u%29%2F3+\"   Yes, it is the same (replace u with n).  Proof complete.\r
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