Observe that
1/1+1/3=4/3;4^2+3^2=5^2
1/3+1/5=8/15;8^2+15^2=17^2
1/5+1/7=12/35;12^2+35^2=37^2
State and prove the generalization suggested by the above examples.

it would seem that you are suggesting that when you add the inverse of consecutive odd integers then resulting numerator and denominator form a pythagorean triple.  This can be show as follows:

1/(2n-1)+1/(2n+1)=
(2n+1+2n-1)/[(2n-1)(2n+1)]=
4n/(4n^2-1)
now
(4n)^2+(4n^2-1)^2=
16n^2+16n^4-8n^2+1=
16n^4+8n^2+1=
(4n^2+1)^2

Posted by Daniel on 2012-12-18 11:13:23