general term of a given sequence a sequence is given thus : $$1 , \frac{1}{2} , 2 , \frac{1}{3} , 3 , \frac{1}{4} , 4 ,\frac{1}{5} ,.....$$
I derived a formula for the general term of this sequence as :
$$X_n = \left[\frac{(n+2^{[(-1)^n + 1]/2} )}{2}\right]^{(-1)^{(n+1)}}$$
but the answer in the book is given thus :
$$X_n = \frac{n+1}{4}[1-(-1)^n] + \frac{ 1}{n+2}[1+ (-1)^n]$$
with $n= 1,2,3,\ldots$
My question is that although both formulas are giving correct terms , I don't see that one is reducible to other one, can a general term be expressed by two or more different forms that are irreducible to each other or are they related to each other in some way?

It is true that
$$(\frac{n+2^{\frac{(-1)^n+1}{2}}}{2})^{(-1)^{n+1}}\neq\frac{n+1}{4}(1-(-1)^n)+\frac{1}{n+2}(1+(-1)^n)$$
But $n\in\mathbb{N}$. Try flooring the $n$ - expressions return equal values. Both answers are correct.
