Prove by mathematical induction that `sum_(r=0)^(n)r^(n)C_(r)=n.2^(n-1), forall n in N`.

Explanation

To prove the statement \( \sum_{r=0}^{n} r \binom{n}{r} = n \cdot 2^{n-1} \) for all \( n \in \mathbb{N} \) using mathematical induction, we will follow these steps:

### Step 1: Base Case
We first check the base case when \( n = 1 \).

**Left-hand side (LHS):**
\[
\sum_{r=0}^{1} r \binom{1}{r} = 0 \cdot \binom{1}{0} + 1 \cdot \binom{1}{1} = 0 + 1 = 1
...