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The expression given is:

\[
\sum_{n=2}^{12} \left(-\frac{1}{2}n + 3\right)
\]

This is a summation from \(n = 2\) to \(n = 12\) of the expression \(-\frac{1}{2}n + 3\).

Let's calculate this step by step:

1. First, we'll calculate the expression \(-\frac{1}{2}n + 3\) for each value of \(n\) from 2 to 12.
2. Then we'll sum up all these values.

### Step 1: Calculate the expression for each \(n\)
- For \(n = 2\): \(-\frac{1}{2}(2) + 3 = -1 + 3 = 2\)
- For \(n = 3\): \(-\frac{1}{2}(3) + 3 = -\frac{3}{2} + 3 = 1.5\)
- For \(n = 4\): \(-\frac{1}{2}(4) + 3 = -2 + 3 = 1\)
- For \(n = 5\): \(-\frac{1}{2}(5) + 3 = -2.5 + 3 = 0.5\)
- For \(n = 6\): \(-\frac{1}{2}(6) + 3 = -3 + 3 = 0\)
- For \(n = 7\): \(-\frac{1}{2}(7) + 3 = -3.5 + 3 = -0.5\)
- For \(n = 8\): \(-\frac{1}{2}(8) + 3 = -4 + 3 = -1\)
- For \(n = 9\): \(-\frac{1}{2}(9) + 3 = -4.5 + 3 = -1.5\)
- For \(n = 10\): \(-\frac{1}{2}(10) + 3 = -5 + 3 = -2\)
- For \(n = 11\): \(-\frac{1}{2}(11) + 3 = -5.5 + 3 = -2.5\)
- For \(n = 12\): \(-\frac{1}{2}(12) + 3 = -6 + 3 = -3\)

### Step 2: Sum the results

\[
2 + 1.5 + 1 + 0.5 + 0 + (-0.5) + (-1) + (-1.5) + (-2) + (-2.5) + (-3)
\]

Now, let's add these up:

\[
2 + 1.5 + 1 + 0.5 + 0 - 0.5 - 1 - 1.5 - 2 - 2.5 - 3 = -6.5
\]

### Final Answer:
The value of the summation is \(-6.5\).

Would you like further details or have any questions?

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**Related Questions:**

1. How would the result change if the limits of the summation were from \( n = 1 \) to \( n = 12 \)?
2. Can you express this summation as a closed-form formula without performing the sum directly?
3. How does the value of the summation change if we modify the expression inside the summation?
4. What would be the result if we summed the expression from \( n = 2 \) to \( n = 20 \)?
5. Can you find a similar summation with a different linear expression, say \( 2n - 5 \)?

**Tip:** When dealing with summations, it's often helpful to compute the first few terms to identify any patterns or simplify the expression if possible.

Calculate the summation \(\sum_{n=2}^{12} \left(-\frac{1}{2}n + 3\right)\).

Learn how to calculate the summation \(\sum_{n=2}^{12} \left(-\frac{1}{2}n + 3\right)\) with a detailed step-by-step solution. This problem involves key algebraic concepts, including summation and linear equations, and is suitable for students in Grades 9-11.

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