If `sum_(i=1)^(7) i^(2)x_(i) = 1` and `sum_(i=1)^(7)(i+1)^(2) x_(i) = 12` and `sum_(i=1)^(7)(i+2)^(2)x_(i) = 123` then find the value of `sum_(i=1)^(7)(i+3)^(2)x_(i)"____"`

Answer

To solve the problem, we need to find the value of \( \sum_{i=1}^{7} (i+3)^2 x_i \) given the following equations:

1. \( \sum_{i=1}^{7} i^2 x_i = 1 \)
2. \( \sum_{i=1}^{7} (i+1)^2 x_i = 12 \)
3. \( \sum_{i=1}^{7} (i+2)^2 x_i = 123 \)

### Step 1: Expand \( \sum_{i=1}^{7} (i+3)^2 x_i \)

Using the identity \( (a+b)^2 = a^2 + 2ab + b^2 \), we can expand \( (i+3)^2 \):

\[ (i+3)^2 = i^2 + 6i + 9 \]

Thus, we can write:

\[ \sum_{i=1}^{7} (i+3)^2 x_i = \sum_{i=1}^{7} (i^2 + 6i + 9) x_i \]

This can be separated into three sums:

\[ \sum_{i=1}^{7} (i+3)^2 x_i = \sum_{i=1}^{7} i^2 x_i + 6 \sum_{i=1}^{7} i x_i + 9 \sum_{i=1}^{7} x_i \]

### Step 2: Substitute known values

We know from the problem statement:

- \( \sum_{i=1}^{7} i^2 x_i = 1 \)

Let’s denote:

- \( S = \sum_{i=1}^{7} x_i \)
- \( T = \sum_{i=1}^{7} i x_i \)

So we can rewrite our expression as:

\[ \sum_{i=1}^{7} (i+3)^2 x_i = 1 + 6T + 9S \]

### Step 3: Find \( T \) and \( S \)

Next, we will find \( S \) and \( T \) using the other two equations.

#### From \( \sum_{i=1}^{7} (i+1)^2 x_i = 12 \):

Expanding \( (i+1)^2 \):

\[ (i+1)^2 = i^2 + 2i + 1 \]

Thus,

\[ \sum_{i=1}^{7} (i+1)^2 x_i = \sum_{i=1}^{7} i^2 x_i + 2 \sum_{i=1}^{7} i x_i + \sum_{i=1}^{7} x_i \]

Substituting the known values:

\[ 12 = 1 + 2T + S \]

Rearranging gives:

\[ 2T + S = 11 \quad \text{(Equation 1)} \]

#### From \( \sum_{i=1}^{7} (i+2)^2 x_i = 123 \):

Expanding \( (i+2)^2 \):

\[ (i+2)^2 = i^2 + 4i + 4 \]

Thus,

\[ \sum_{i=1}^{7} (i+2)^2 x_i = \sum_{i=1}^{7} i^2 x_i + 4 \sum_{i=1}^{7} i x_i + 4 \sum_{i=1}^{7} x_i \]

Substituting the known values:

\[ 123 = 1 + 4T + 4S \]

Rearranging gives:

\[ 4T + 4S = 122 \quad \text{or} \quad T + S = 30.5 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations

Now we have a system of equations:

1. \( 2T + S = 11 \)
2. \( T + S = 30.5 \)

Subtract Equation 1 from Equation 2:

\[ (T + S) - (2T + S) = 30.5 - 11 \]

This simplifies to:

\[ -T = 19.5 \quad \Rightarrow \quad T = -19.5 \]

Substituting \( T \) back into Equation 2:

\[ -19.5 + S = 30.5 \quad \Rightarrow \quad S = 50 \]

### Step 5: Substitute \( T \) and \( S \) back into the expression

Now we have \( S = 50 \) and \( T = -19.5 \). Substitute these values back into our expression for \( \sum_{i=1}^{7} (i+3)^2 x_i \):

\[ \sum_{i=1}^{7} (i+3)^2 x_i = 1 + 6(-19.5) + 9(50) \]

Calculating this gives:

\[ = 1 - 117 + 450 \]
\[ = 334 \]

### Final Answer

Thus, the value of \( \sum_{i=1}^{7} (i+3)^2 x_i \) is \( \boxed{334} \).