This is a wicked problem!
I examined the denominators,Find a formula for this sequence: .
I considered the denominators for odd and for even
. . If is odd, add 1 and divide by 2: . .
. . If is even, divide by 2 and add 2: . .
How can we alternate between the two formulas: . and  ?
. . Multiply  by: . multiply  by: . and add the results.
We have: .
. . This simplifies to: .
. . Therefore: .