SOLUTION: Use mathematical induction to prove that: n ∑ r^2/{(2r-1)(2r+1)}= n(n+1)/(2(2n+1)) r=1 Thanks!!

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Question 941989: Use mathematical induction to prove that:
n
∑ r^2/{(2r-1)(2r+1)}= n(n+1)/(2(2n+1))
r=1
Thanks!!
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
We first want to prove that
if equation (1) were true for some value of n,

(1) then it would also be true when we replace n by n+1. That is,
we want to show that if (1) holds for some value of n, then we
can erase the ??? below and replace it with an equal sign:

which simplifies to:
(2)
-----------------------
So we add the (n+1)st term to both sides of (1)

and simplify. The left side is just the sum to n+1 terms instead of n
terms:

Get an LCD by multiplying the first fraction on the right by and the second by


Combine numerators over the LCD

Factor out (n+1) on top




Now cancel the (2n+1)'s

That is the same as (2)
All that's left is to show that (1) is true for n=1





So since it works for n=1, it works for n=2. Then
since it works for n=2, it works for n=3, then for n=4
and so on forever for all values on n.
Edwin
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