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Let's solve each summation step by step based on the given values of \( X_1 = 3 \), \( X_2 = 4 \), \( X_3 = 7 \), and \( X_4 = 12 \).

### 1. \( \sum_{i=1}^{2} (X_i + 2) \)

This means we will calculate for \( i = 1 \) and \( i = 2 \):

\[
X_1 + 2 = 3 + 2 = 5
\]
\[
X_2 + 2 = 4 + 2 = 6
\]

Now sum the results:

\[
5 + 6 = 11
\]

**Answer: 11**

---

### 2. \( \sum_{i=1}^{3} (X_i - 3) \)

This summation runs from \( i = 1 \) to \( i = 3 \):

\[
X_1 - 3 = 3 - 3 = 0
\]
\[
X_2 - 3 = 4 - 3 = 1
\]
\[
X_3 - 3 = 7 - 3 = 4
\]

Now sum the results:

\[
0 + 1 + 4 = 5
\]

**Answer: 5**

---

### 3. \( \sum_{i=1}^{4} (2X_i) + 2 \)

This summation involves multiplying each \( X_i \) by 2, then adding 2 at the end:

\[
2X_1 = 2 \times 3 = 6
\]
\[
2X_2 = 2 \times 4 = 8
\]
\[
2X_3 = 2 \times 7 = 14
\]
\[
2X_4 = 2 \times 12 = 24
\]

Now sum these results:

\[
6 + 8 + 14 + 24 = 52
\]

Finally, add 2:

\[
52 + 2 = 54
\]

**Answer: 54**

---

### 4. \( \sum_{i=1}^{2} \left(\frac{X_i}{4}\right) \)

This summation divides each \( X_i \) by 4 for \( i = 1 \) and \( i = 2 \):

\[
\frac{X_1}{4} = \frac{3}{4} = 0.75
\]
\[
\frac{X_2}{4} = \frac{4}{4} = 1
\]

Now sum the results:

\[
0.75 + 1 = 1.75
\]

**Answer: 1.75**

---

If you would like more details or explanations, feel free to ask!

### Related questions:
1. How do you expand the summation notation into individual terms?
2. What is the importance of the summation symbol in mathematics?
3. How would the result change if \( X_3 \) were different in any of the problems?
4. Can summations be used to find the average of a series of numbers? How?
5. What are some common applications of summation in real-world problems?

### Tip:
When working with summations, always ensure to carefully expand each term based on the limits of the summation to avoid errors.

Given the values X1 = 3, X2 = 4, X3 = 7, X4 = 12, solve the summations: ∑(X_i + 2) for i = 1 to 2, ∑(X_i - 3) for i = 1 to 3, ∑(2X_i) + 2 for i = 1 to 4, ∑(X_i / 4) for i = 1 to 2.

This problem involves solving multiple summations using given values of X1 = 3, X2 = 4, X3 = 7, and X4 = 12. Learn step-by-step how to calculate each summation and understand the core concepts of algebraic summation.

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