document.write( "Question 941989: Use mathematical induction to prove that:
\n" ); document.write( "n
\n" ); document.write( "∑ r^2/{(2r-1)(2r+1)}= n(n+1)/(2(2n+1))
\n" ); document.write( "r=1\r
\n" ); document.write( "\n" ); document.write( "Thanks!! \n" ); document.write( "

Algebra.Com's Answer #576391 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!
We first want to prove that
\n" ); document.write( "if equation (1) were true for some value of n,\r
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\n" ); document.write( "\n" ); document.write( "(1)

$\"%22%22=%22%22\"$ $\"n%28n%2B1%29%2F%282%282n%2B1%29%29\"$ then it would also be true when we replace n by n+1. That is,
\n" ); document.write( "we want to show that if (1) holds for some value of n, then we
\n" ); document.write( "can erase the ??? below and replace it with an equal sign:\r
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$\"%22%3F%3F%3F%22\"$

\r
\n" ); document.write( "\n" ); document.write( "which simplifies to:\r
\n" ); document.write( "\n" ); document.write( "(2)

$\"%22%3F%3F%3F%22\"$ $\"%28n%2B1%29%28n%2B2%29%2F%282%282n%2B3%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( "-----------------------\r
\n" ); document.write( "\n" ); document.write( "So we add the (n+1)st term to both sides of (1)\r
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$\"%22%22%2B%22%22\"$ $\"%28n%2B1%29%5E2%2F%28%282%28n%2B1%29%5E%22%22-1%29%282%28n%2B1%29%5E%22%22%2B1%29%29\"$ $\"%22%22=%22%22\"$ $\"n%28n%2B1%29%5E%22%22%2F%282%282n%2B1%29%5E%22%22%29\"$ $\"%22%22%2B%22%22\"$ $\"%28n%2B1%29%5E2%2F%28%282%28n%2B1%29%5E%22%22-1%29%282%28n%2B1%29%5E%22%22%2B1%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( "and simplify. The left side is just the sum to n+1 terms instead of n
\n" ); document.write( "terms:\r
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$\"%22%22=%22%22\"$ $\"n%28n%2B1%29%5E%22%22%2F%282%282n%2B1%29%5E%22%22%29\"$ $\"%22%22%2B%22%22\"$ $\"%28n%2B1%29%5E2%2F%282n%2B1%29%282n%2B3%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( "Get an LCD by multiplying the first fraction on the right by $\"%28%282n%2B3%29%29%2F%28%282n%2B3%29%29\"$ and the second by $\"2%2F2\"$ \r
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$\"%22%22=%22%22\"$ $\"n%28n%2B1%29%5E%22%22%282n%2B3%29%2F%282%282n%2B1%29%5E%22%22%282n%2B3%29%29\"$ $\"%22%22%2B%22%22\"$ $\"2%28n%2B1%29%5E2%2F2%282n%2B1%29%282n%2B3%29%29\"$ \r
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$\"%22%22=%22%22\"$

\r
\n" ); document.write( "\n" ); document.write( "Factor out (n+1) on top\r
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$\"%22%22=%22%22\"$

\r
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$\"%22%22=%22%22\"$ $\"%28%28n%2B1%29%282n%5E2%2B3n%2B2n%2B2%29%29%2F%282%282n%2B1%29%5E%22%22%282n%2B3%29%29\"$ \r
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$\"%22%22=%22%22\"$ $\"%28%28n%2B1%29%282n%5E2%2B5n%2B2%29%29%2F%282%282n%2B1%29%5E%22%22%282n%2B3%29%29\"$ \r
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$\"%22%22=%22%22\"$ $\"%28%28n%2B1%29%282n%2B1%29%28n%2B2%29%5E%22%22%29%2F%282%282n%2B1%29%5E%22%22%282n%2B3%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( "Now cancel the (2n+1)'s\r
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$\"%22%22=%22%22\"$ $\"%28%28n%2B1%29%28n%2B2%29%5E%22%22%29%2F%282%282n%2B3%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( "That is the same as (2)\r
\n" ); document.write( "\n" ); document.write( "All that's left is to show that (1) is true for n=1\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"1%5E2%2F%28%282%281%29%5E%22%22-1%29%282%281%29%5E%22%22%2B1%29++++++++%29\"$ $\"%22%22=%22%22\"$ $\"1%281%2B1%29%2F%282%282%281%29%5E%22%22%2B1%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"1%2F%28%282-1%29%282%2B1%29++++++++%29\"$ $\"%22%22=%22%22\"$ $\"1%282%29%2F%282%282%2B1%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"1%2F%28%281%29%283%29++++++++%29\"$ $\"%22%22=%22%22\"$ $\"2%2F%282%283%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"1%2F3\"$ $\"%22%22=%22%22\"$ $\"1%2F3\"$ \r
\n" ); document.write( "\n" ); document.write( "So since it works for n=1, it works for n=2. Then
\n" ); document.write( "since it works for n=2, it works for n=3, then for n=4
\n" ); document.write( "and so on forever for all values on n.\r
\n" ); document.write( "\n" ); document.write( "Edwin \n" ); document.write( "

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