The expression provided is a summation:

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

This means that you need to evaluate the sum of the expression $-\frac{1}{2}n + 3$ for each integer $n$ from 2 to 12.

Let's compute this step by step.

Step 1: Evaluate the expression for each value of $n$ from 2 to 12.

For $n = 2$: $-\frac{1}{2}(2) + 3 = -1 + 3 = 2$

For $n = 3$: $-\frac{1}{2}(3) + 3 = -\frac{3}{2} + 3 = \frac{3}{2} = 1.5$

For $n = 4$: $-\frac{1}{2}(4) + 3 = -2 + 3 = 1$

For $n = 5$: $-\frac{1}{2}(5) + 3 = -\frac{5}{2} + 3 = 0.5$

For $n = 6$: $-\frac{1}{2}(6) + 3 = -3 + 3 = 0$

For $n = 7$: $-\frac{1}{2}(7) + 3 = -\frac{7}{2} + 3 = -0.5$

For $n = 8$: $-\frac{1}{2}(8) + 3 = -4 + 3 = -1$

For $n = 9$: $-\frac{1}{2}(9) + 3 = -\frac{9}{2} + 3 = -1.5$

For $n = 10$: $-\frac{1}{2}(10) + 3 = -5 + 3 = -2$

For $n = 11$: $-\frac{1}{2}(11) + 3 = -\frac{11}{2} + 3 = -2.5$

For $n = 12$: $-\frac{1}{2}(12) + 3 = -6 + 3 = -3$

Step 2: Sum these values.

$2 + 1.5 + 1 + 0.5 + 0 - 0.5 - 1 - 1.5 - 2 - 2.5 - 3 = -6.5$

So, the value of the summation is $-6.5$.

Would you like more details, or do you have any questions?

Here are five related questions to deepen your understanding:

1. What are the general properties of summation notation?
2. How does changing the upper and lower bounds of a summation affect its value?
3. How can summation be applied to sequences, and what patterns might emerge?
4. What is the significance of arithmetic progressions in summation problems?
5. How can you use calculus to approximate or evaluate summations of continuous functions?

Tip: When working with summations, break down the problem into smaller steps, such as evaluating each term first, then summing them together. This helps reduce errors and makes the process clearer.