For all positive integers `n ,` show that `\ ^(2n)C_n+\ ^(2n)C_(n-1)=1/2(\ ^(2n+2)C_(n+1))` .

Explanation

To prove the identity \( \binom{2n}{n} + \binom{2n}{n-1} = \frac{1}{2} \binom{2n+2}{n+1} \) for all positive integers \( n \), we will start by analyzing the left-hand side (LHS) and the right-hand side (RHS) of the equation.

### Step 1: Start with the LHS
The left-hand side is given by:
\[ \text{LHS} = \binom{2n}{n} + \binom{2n}{n-1} \]

...