https://file.aiquickdraw.com/mathbot/file/1726897757618uq82wts6.jpg

Let's break down and solve the questions from the image:

### Questions 1–3: 

**Given dataLet's break down and solve the questions from the image:

### Questions 1–3: 

**Given **
- \(x_1 = 22\), \(x_2 = 18\), \(x_3 = 25\), \(x_4 = 20\), \(x_5 = 12\), \(x_6 = 17\), \(x_7 = 25\), \(x_8 = 26\), \(x_9 = 16\), \(x_{10} = 20\)

1. **Evaluate** \( \sum_{i=1}^{10} x_i \)  
   This is the sum of the values from \(x_1\) to \(x_{10}\).

   \[
   \sum_{i=1}^{10} x_i = 22 + 18 + 25 + 20 + 12 + 17 + 25 + 26 + 16 + 20
   \]
   Calculating the sum gives:  
   \[
   \sum_{i=1}^{10} x_i = 201
   \]
   **Answer: D. 201**

2. **Evaluate** \( \sum_{i=4}^{6} x_i^2 \)  
   This is the sum of the squares of \(x_4\), \(x_5\), and \(x_6\).

   \[
   x_4^2 = 20^2 = 400,\quad x_5^2 = 12^2 = 144,\quad x_6^2 = 17^2 = 289
   \]
   \[
   \sum_{i=4}^{6} x_i^2 = 400 + 144 + 289 = 833
   \]
   **Answer: B. 833**

3. **Evaluate** \( \sum_{i=3}^{7} 4x_i \)  
   This is the sum of four times \(x_i\) for \(i = 3\) to \(i = 7\).

   \[
   4x_3 = 4 \times 25 = 100,\quad 4x_4 = 4 \times 20 = 80,\quad 4x_5 = 4 \times 12 = 48,\quad 4x_6 = 4 \times 17 = 68,\quad 4x_7 = 4 \times 25 = 100
   \]
   \[
   \sum_{i=3}^{7} 4x_i = 100 + 80 + 48 + 68 + 100 = 396
   \]
   **Answer: A. 396**

### Questions 4–6: 

**Given:**
- \(a_1 = 23\) and \( \sum_{i=1}^{12} a_i = 947\)

4. **Find** \( \sum_{i=1}^{12} 10a_i \)  
   This is 10 times the sum of \(a_1\) to \(a_{12}\).

   \[
   \sum_{i=1}^{12} 10a_i = 10 \times \sum_{i=1}^{12} a_i = 10 \times 947 = 9470
   \]
   **Answer: B. 9470**

5. **Find** \( \sum_{i=1}^{12} (5a_i - 24) \)  
   This can be split into two sums:  
   \[
   \sum_{i=1}^{12} (5a_i - 24) = 5\sum_{i=1}^{12} a_i - 12 \times 24
   \]
   \[
   = 5 \times 947 - 288 = 4735 - 288 = 4447
   \]
   **Answer: B. 4447**

6. **Find** \( \sum_{i=2}^{12} (2a_i + 5) \)  
   This can be split into two sums:  
   \[
   \sum_{i=2}^{12} (2a_i + 5) = 2 \sum_{i=2}^{12} a_i + 11 \times 5
   \]
   Since \( \sum_{i=2}^{12} a_i = \sum_{i=1}^{12} a_i - a_1 = 947 - 23 = 924 \):
   \[
   2 \times 924 + 55 = 1848 + 55 = 1903
   \]
   **Answer: D. 1903**

If you would like to go further with the remaining problems or any additional details, let me know!

### Expanding Questions:
1. What is the formula for calculating the sum of squares \( \sum_{i=1}^{n} x_i^2 \)?
2. How does the inclusion of constants (like 5 in question 5) affect summation?
3. How is the arithmetic series sum calculated when there is a linear transformation inside the sum (e.g., \(2a_i + 5\))?
4. What patterns or simplifications can be used when working with sums involving constants and variables?
5. How would you calculate a sum involving multiple nested sums?

### Tip:
When evaluating complex summations, try breaking them into simpler, more manageable parts—this can make calculations faster and more accurate!

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