Question

he value of 1/3*5+1/5*7+1/7*9+....+1/23*25 is

Answer 1

It may help to rewrite the terms a bit:

Numbering the terms:

1: 1/(3*5)
2: 1/(5*7)
3: 1/(7*9)
etc

We can refer to the terms by their sequence numbers

\(\dfrac{1}{(2(1)+1)\cdot (2(1) + 3)} + \dfrac{1}{(2(2)+1)\cdot (2(2) + 3)} + ... etc\)

Or generally, we have for n = 1, 2, 3, ... 11, just \(\dfrac{1}{(2n+1)\cdot (2n + 3)}\)

That doesn't do much until you find that: \(\dfrac{1}{(2n+1)\cdot (2n + 3)} = \dfrac{1/2}{2n+1} - \dfrac{1/2}{2n+3}\).

If you write all your terms like that last expression, something wonderful will occur.