If the value of
` (n + 2) . ""^(n)C_(0) *2^(n+1) - (n+1) * ""^(n)C_(1)*2^(n) + n* ""^(n)C_(2) * 2^(n-1) -...`
is equal to `k(n +1)` , the value of k is .

Answer

To solve the problem, we need to evaluate the expression given and find the value of \( k \) such that the expression equals \( k(n + 1) \).

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression given is:
\[ (n + 2) \cdot \binom{n}{0} \cdot 2^{n+1} - (n + 1) \cdot \binom{n}{1} \cdot 2^n + n \cdot \binom{n}{2} \cdot 2^{n-1} - \ldots \]
This is a series involving binomial coefficients and powers of 2.

2. **Substituting Values of \( n \)**:
To find a pattern, we will substitute small integer values for \( n \).

- **For \( n = 1 \)**:
\[ (1 + 2) \cdot \binom{1}{0} \cdot 2^{1+1} - (1 + 1) \cdot \binom{1}{1} \cdot 2^1 + 1 \cdot \binom{1}{2} \cdot 2^{1-1} \]
Simplifying this:
\[ 3 \cdot 1 \cdot 4 - 2 \cdot 1 \cdot 2 + 1 \cdot 0 \cdot 1 = 12 - 4 + 0 = 8 \]

3. **Finding the General Form**:
We observe that for \( n = 1 \), the expression evaluates to 8. We can express this as:
\[ 8 = 4 \cdot (1 + 1) = 4 \cdot 2 \]

4. **Generalizing for \( n \)**:
From our calculations, we can see that the expression seems to yield:
\[ 4(n + 1) \]
for \( n = 1 \).

5. **Comparing with the Given Expression**:
We are given that the expression equals \( k(n + 1) \). From our findings:
\[ 4(n + 1) = k(n + 1) \]
This implies that \( k = 4 \).

### Conclusion:
Thus, the value of \( k \) is:
\[ \boxed{4} \]