Evaluate the summation below.
sum, from, m, equals, 4, to, 8, of, left parenthesis, minus, 7, m, plus, m, squared, right parenthesis
m=4
∑
8
​
 (−7m+m 
2
 )

We are given the summation:

\[
\sum_{m=4}^{8} (-7m + m^2)
\]

This means we need to evaluate the expression \( (-7m + m^2) \) for each value of \( m \) from 4 to 8, and then sum the results.

Let's compute the values step by step:

1. For \( m = 4 \):
\[
-7(4) + 4^2 = -28 + 16 = -12
\]

2. For \( m = 5 \):
\[
-7(5) + 5^2 = -35 + 25 = -10
\]

3. For \( m = 6 \):
\[
-7(6) + 6^2 = -42 + 36 = -6
\]

4. For \( m = 7 \):
\[
-7(7) + 7^2 = -49 + 49 = 0
\]

5. For \( m = 8 \):
\[
-7(8) + 8^2 = -56 + 64 = 8
\]

Now, summing these results:

\[
-12 + (-10) + (-6) + 0 + 8 = -12 - 10 - 6 + 0 + 8 = -20
\]

Thus, the value of the summation is:

\[
\boxed{-20}
\]

Would you like more details or have any questions?

---

Here are 5 related questions to explore:
1. What is the general formula for evaluating arithmetic-geometric sums?
2. How does the summation change if the limits are extended?
3. How do different summation bounds affect the total?
4. What happens if we add a constant to the expression inside the summation?
5. Can this sum be computed using a closed-form formula?

**Tip:** Summations often simplify by identifying patterns or using formulas like arithmetic or geometric series.

Evaluate the summation: sum, from, m = 4 to 8, of (-7m + m^2).

Learn how to evaluate the summation of the expression (-7m + m^2) for values of m from 4 to 8. This problem involves summation techniques and quadratic functions, providing a step-by-step breakdown of the solution. Suitable for high school students (Grades 9-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation