Suppose `sum_(r=0)^2023r^2"^(2023)C_r=2023 xx alpha xx 2^(2022)` Then the value of `alpha` is.

Answer

To solve the problem, we need to evaluate the expression:

\[ \sum_{r=0}^{2023} r^2 \binom{2023}{r} \]

and show that it equals \(2023 \times \alpha \times 2^{2022}\).

### Step 1: Use the identity for \( r^2 \)

We can express \( r^2 \) in terms of binomial coefficients. The identity we can use is:

\[ r^2 = r(r-1) + r \]

Thus, we can rewrite the sum as:

\[ \sum_{r=0}^{2023} r^2 \binom{2023}{r} = \sum_{r=0}^{2023} (r(r-1) + r) \binom{2023}{r} \]

### Step 2: Break the sum into two parts

Now, we can separate the sum into two parts:

\[ \sum_{r=0}^{2023} r^2 \binom{2023}{r} = \sum_{r=0}^{2023} r(r-1) \binom{2023}{r} + \sum_{r=0}^{2023} r \binom{2023}{r} \]

### Step 3: Evaluate the first sum

For the first sum, we can use the identity:

\[ r(r-1) \binom{n}{r} = n(n-1) \binom{n-2}{r-2} \]

Thus, we have:

\[ \sum_{r=0}^{2023} r(r-1) \binom{2023}{r} = 2023 \times 2022 \sum_{r=2}^{2023} \binom{2021}{r-2} = 2023 \times 2022 \cdot 2^{2021} \]

### Step 4: Evaluate the second sum

For the second sum, we can use the identity:

\[ \sum_{r=0}^{n} r \binom{n}{r} = n \cdot 2^{n-1} \]

Thus, we have:

\[ \sum_{r=0}^{2023} r \binom{2023}{r} = 2023 \cdot 2^{2022} \]

### Step 5: Combine the results

Now we can combine both parts:

\[ \sum_{r=0}^{2023} r^2 \binom{2023}{r} = 2023 \times 2022 \cdot 2^{2021} + 2023 \cdot 2^{2022} \]

Factoring out \(2023\):

\[ = 2023 \left( 2022 \cdot 2^{2021} + 2^{2022} \right) \]

### Step 6: Simplify the expression

Notice that \(2^{2022} = 2 \cdot 2^{2021}\), so we can rewrite:

\[ = 2023 \left( 2022 \cdot 2^{2021} + 2 \cdot 2^{2021} \right) = 2023 \left( (2022 + 2) \cdot 2^{2021} \right) = 2023 \cdot 2024 \cdot 2^{2021} \]

### Step 7: Relate to the original equation

Now we have:

\[ 2023 \cdot 2024 \cdot 2^{2021} = 2023 \cdot \alpha \cdot 2^{2022} \]

Dividing both sides by \(2023 \cdot 2^{2021}\):

\[ 2024 = \alpha \cdot 2 \]

### Step 8: Solve for \(\alpha\)

Thus, we can solve for \(\alpha\):

\[ \alpha = \frac{2024}{2} = 1012 \]

### Final Answer

The value of \(\alpha\) is:

\[ \boxed{1012} \]