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!