If `f(x) = (a^(x))/(a^(x) + sqrt(a)), a gt 0` prove that
(i) `f(x) + f(1-x) = 1`
(ii) `underset(k=1)overset(2n-1)sum 2f ((k)/(2n)) = 2n-1`
(iii) `underset(k=1)overset(2n)sum 2f((k)/(2n+1)) = 2n`

Explanation

To solve the given problem, we will tackle each part step-by-step.

### Given:
\[ f(x) = \frac{a^x}{a^x + \sqrt{a}} \]

### (i) Prove that \( f(x) + f(1-x) = 1 \)

1. **Calculate \( f(1-x) \)**:
...