document.write( "Question 972088: Assume 1^2 + 2^2 + 3^2 +... n^2 = (n(n+1)(2n+1))/6 is true for all positive integers n. If we replace the right hand side with ((n+2)(n+3)(2n+5))/6 , what term(s) do we add to the left hand side?\r
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Algebra.Com's Answer #594476 by Edwin McCravy(20056) $\"\"$ $\"About$

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Assume 1^2 + 2^2 + 3^2 +... n^2 = (n(n+1)(2n+1))/6 is true for all positive integers n. If we replace the right hand side with ((n+2)(n+3)(2n+5))/6 , what term(s) do we add to the left hand side?\r
\n" ); document.write( "\n" ); document.write( "I believe the correct answer is (n+1)^2 but I want to double check
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document.write( "I'm afraid that's not all but only part of what is added to the left side.\r\n" );
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document.write( "To find out what we have added to the left hand side when we have replaced\r\n" );
document.write( "the right side with  we compare them:\r\n" );
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document.write( "We see that the first factor on the left numerator is n, while the first\r\n" );
document.write( "factor on the right numerator is (n+2), so that makes us suspect that n \r\n" );
document.write( "has been replaced by (n+2).  \r\n" );
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document.write( "We check to see if that is the case with the second factor on the left. We take\r\n" );
document.write( "(n+1) and replace n by n+2 in it and we get (n+2+1) or (n+3) which is the second\r\n" );
document.write( "factor on the right.  So far so good.\r\n" );
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document.write( "We only need to show that if we replace n by n+2 in the third factor on the left\r\n" );
document.write( "that we will get the third factor on the right (2n+5).  We show that by \r\n" );
document.write( "replacing n by n+2 in (2n+1):\r\n" );
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document.write( "(2(n+2)+1) = (2n+4+1) = (2n+5)\r\n" );
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document.write( "Now we know that the right side has been replaced by (n+2) throughout.\r\n" );
document.write( "Therefore we know that the sum on the left, which is the sum up through n terms,\r\n" );
document.write( "is now to be carried up to (n+2) terms, which is 2 more terms, so now we\r\n" );
document.write( "have\r\n" );
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document.write( "Therefore two more terms  have been added to the left,\r\n" );
document.write( "not just the one term that you were thinking was added.\r\n" );
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document.write( "Edwin

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