https://r2.erweima.ai/mathbot/image/1725929962673g8w5c8sa.jpg

Let's break down the summation exercises shown in the image step by step.

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

#### Part 1: Developing the summations using the given values

1. **Summation a)** 
   \[
   \sum_{i=1}^{4} \left( x_i + a \right)
   \]
   Using the values of \( x_1, x_2, x_3, x_4 \), we compute:
   \[
   \sum_{i=1}^{4} \left( x_i + a \right) = (x_1 + a) + (x_2 + a) + (x_3 + a) + (x_4 + a)
   \]
   Substituting the values:
   \[
   (7 + 10) + (8 + 10) + (10 + 10) + (6 + 10) = 17 + 18 + 20 + 16 = 71
   \]

2. **Summation b)** 
   \[
   \sum_{i=2}^{84} a x_i
   \]
   Since the range is from \(i=2\) to \(i=84\), but we only have \( x_2, x_3, x_4 \), we use those values:
   \[
   a x_2 + a x_3 + a x_4 = 10 \cdot 8 + 10 \cdot 10 + 10 \cdot 6 = 80 + 100 + 60 = 240
   \]

3. **Summation c)**
   \[
   \sum_{i=26}^{26} a
   \]
   Since the sum only considers \( i = 26 \), it is simply \( a = 10 \).

---

#### Part 2: Expressing the sums using summation notation

a) The sum provided is:
   \[
   (84 x_1 - 26 y_1) + (84 x_2 - 26 y_2) + (84 x_3 - 26 y_3) + \dots
   \]
   This can be expressed as:
   \[
   \sum_{i=1}^{n} (84 x_i - 26 y_i)
   \]
   where \( n \) represents the number of terms.

b) The second sum involves another similar sum, likely expressible in a similar summation notation, but from the image it is not entirely clear.

---

### Would you like further details on any part of the solution, or assistance with a different summation problem? Here are some related questions that might help deepen understanding:

1. How would the result change if we added more values for \(x_i\)?
2. Can you generalize the second summation in part b) for a broader range of \( i \)?
3. What happens if the constant \(a\) is negative instead of positive?
4. Can you express the sum in part 2a) for \(n\) terms instead of using a fixed number of terms?
5. How would you calculate the sum if the starting point was different (e.g., \(i = 3\) instead of \(i = 2\))?

**Tip:** When dealing with summations, carefully check the bounds of the index \(i\) and substitute all known values to simplify the sum.

Desarrollar las siguientes sumatorias utilizando los valores anteriores y calcular en los casos posibles: a) \( \sum_{i=1}^{4} \left( x_i + a \right) \), b) \( \sum_{i=2}^{84} a x_i \), c) \( \sum_{i=26}^{26} a \), y expresar las siguientes sumas utilizando el símbolo de la sumatoria: a) \((84x_1 - 26y_1) + (84x_2 - 26y_2) + (84x_3 - 26y_3) + ... \), b) (una segunda suma similar no clara).

Solve summation problems involving constants and variable expressions. Learn step-by-step solutions for summations such as \( \sum_{i=1}^{4} \left( x_i + a \right) \), \( \sum_{i=2}^{84} a x_i \), and more. Understand the application of summation properties and algebraic manipulations. Suitable for students in Grades 9-12.

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