https://r2.erweima.ai/mathbot/image/1726006551231o8140ryd.jpg

Let's go through the problem in detail. The image contains a section on "Summatorias" (Summations) with given values for \(x_1, x_2, x_3, x_4\) and a constant \(a\). Here's how we can break it down:

### Given:
- \( a = 10 \)
- \( x_1 = 7, x_2 = 8, x_3 = 10, x_4 = 6 \)

The task is to calculate the following summations:

1. **Part (a):** 
   \[
   \sum_{i=1}^{4} \left(\frac{4x_i}{x_i + a}\right)
   \]
   We need to calculate this sum using the values of \(x_1, x_2, x_3,\) and \(x_4\), and with \(a = 10\).

2. **Part (b):**
   \[
   \sum_{i=2}^{4} a x_i
   \]
   Here, we sum from \(i = 2\) to \(i = 4\), using \(a = 10\) and the corresponding \(x_i\) values.

3. **Part (c):**
   \[
   \sum_{i=26}^{84} a
   \]
   This is a simple summation of the constant \(a = 10\) over the range from 26 to 84.

Next, there are two expressions that ask to express sums using summation symbols:

- **Expression (a)**:
   \[
   (84x_1 - 26y_1) + (84x_2 - 26y_2) + \dots + (84x_n - 26y_n)
   \]
   We express this as a summation.

- **Expression (b)**:
   \[
   \left(\frac{x_0 - 49}{w_0}\right)^8 + \left(\frac{x_1 - 49}{w_1}\right)^8 + \dots + \left(\frac{x_{65} - 49}{w_{65}}\right)^8
   \]
   Similarly, this expression needs to be written as a sum.

---

Let me start by calculating the results for part (a), (b), and (c) of the summation problems.Here are the results for the summations:

1. **Part (a):**
   \[
   \sum_{i=1}^{4} \left(\frac{4x_i}{x_i + a}\right) = 6.925
   \]
   
2. **Part (b):**
   \[
   \sum_{i=2}^{4} a x_i = 240
   \]

3. **Part (c):**
   \[
   \sum_{i=26}^{84} a = 590
   \]

Next, let's express the two given expressions as summations:

- **Expression (a)**:
   \[
   \sum_{i=1}^{n} (84x_i - 26y_i)
   \]
   This represents a sum over all the \(x_i\) and \(y_i\) terms for \(i = 1\) to \(n\).

- **Expression (b)**:
   \[
   \sum_{i=0}^{65} \left(\frac{x_i - 49}{w_i}\right)^8
   \]
   This is a sum of each term raised to the power of 8, from \(i = 0\) to \(i = 65\).

---

Would you like further details on any of these results?

Here are 5 related questions:
1. How do you generalize the sum of a constant over a range?
2. What is the effect of different values of \(a\) on the summation in part (a)?
3. How would the result change if the range for part (b) were extended to include more terms?
4. How do powers influence the convergence of summations like the one in expression (b)?
5. How could the summation in expression (a) be simplified using matrix notation?

**Tip:** When dealing with summations over a range of indices, it's useful to split the summation into simpler components if needed.

Calculate the given summations with the provided values for x1, x2, x3, x4, and constant a.

Learn how to calculate summations using the given values for x1, x2, x3, x4, and a constant a. This problem focuses on algebraic summations, including different expressions and their calculations. Suitable for high school students in Grades 10-12.

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