https://r2.erweima.ai/mathbot/image/17260031007302p4i7ms0.jpg

Let's break down the summation problems in the image step by step, starting with the values provided:

We are given the following variables:
- \( a = 10 \) (a constant)
- \( x_1 = 7 \), \( x_2 = 8 \), \( x_3 = 10 \), \( x_4 = 6 \)

We are asked to calculate the following summations:

### a) \( \sum_{i=1}^{4} \left( \frac{4x_i}{x_i + a} \right) \)

Using the provided values for \( x_1, x_2, x_3, x_4 \), and \( a = 10 \):

\[
S_a = \frac{4x_1}{x_1 + a} + \frac{4x_2}{x_2 + a} + \frac{4x_3}{x_3 + a} + \frac{4x_4}{x_4 + a}
\]

Substitute the values:

\[
S_a = \frac{4 \times 7}{7 + 10} + \frac{4 \times 8}{8 + 10} + \frac{4 \times 10}{10 + 10} + \frac{4 \times 6}{6 + 10}
\]

This simplifies to:

\[
S_a = \frac{28}{17} + \frac{32}{18} + \frac{40}{20} + \frac{24}{16}
\]

Now compute each term:

\[
S_a = 1.647 + 1.778 + 2 + 1.5
\]

\[
S_a \approx 6.925
\]

### b) \( \sum_{i=2}^{4} a x_i \)

Since \( a = 10 \), we calculate:

\[
S_b = 10 \times x_2 + 10 \times x_3 + 10 \times x_4
\]

Substitute the values for \( x_2, x_3, x_4 \):

\[
S_b = 10 \times 8 + 10 \times 10 + 10 \times 6
\]

\[
S_b = 80 + 100 + 60 = 240
\]

### c) \( \sum_{i=26}^{84} a \)

This is a simple summation of a constant. Since \( a = 10 \), the sum is:

\[
S_c = a \times (84 - 26 + 1) = 10 \times 59 = 590
\]

---

The second part of the problem asks to express the following sums using summation notation:

### a) 
\[
(84x_1 - 26y_1) + (84x_2 - 26y_2) + (84x_3 - 26y_3) + \ldots + (84x_n - 26y_n)
\]

This can be written as:

\[
\sum_{i=1}^{n} (84x_i - 26y_i)
\]

### b)
\[
\left( \frac{x_0 - 49}{w_0} \right)^8 + \left( \frac{x_1 - 49}{w_1} \right)^8 + \ldots + \left( \frac{x_{65} - 49}{w_{65}} \right)^8
\]

This can be written as:

\[
\sum_{i=0}^{65} \left( \frac{x_i - 49}{w_i} \right)^8
\]

---

Would you like more details or have any questions about these solutions?

Here are five related questions that can expand on the topic:
1. What are the rules for summation notation and how do they simplify complex sums?
2. How can summations be applied to real-world problems, like calculating total costs or distances?
3. How can you generalize summations for larger sets of data?
4. What are the applications of constant summations like in part (c) of the question?
5. How can summations involving fractions, like in part (a), be simplified algebraically?

**Tip**: When working with summations, always ensure that you understand the index range (i.e., starting and ending values) to correctly calculate the total sum.

Develop the following summations using the given values and calculate the cases provided in the image.

Learn how to solve complex summation problems involving fractions, constants, and algebraic expressions. This tutorial covers key concepts such as summation notation, fraction simplification, and applying constants in sums. Suitable for high school students in Grades 9-12.

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