x, 17, 3x - y2 - 2 and 3x + y2 - 30 are four consecutive terms of an increasing arithmetic sequence. The sum of the four numbers is divisible by:

Given that \( x \), \( 17 \), \( 3x - y^2 - 2 \), and \( 3x + y^2 - 30 \) are four consecutive terms of an increasing arithmetic sequence, we can solve this problem step by step.

### Step 1: Understand the conditions of an arithmetic sequence
For any arithmetic sequence, the difference between consecutive terms is constant. If \( a_1 \), \( a_2 \), \( a_3 \), and \( a_4 \) are the four terms, we have:
\[
a_2 - a_1 = a_3 - a_2 = a_4 - a_3
\]

### Step 2: Set up the equations
Using the given terms \( x \), \( 17 \), \( 3x - y^2 - 2 \), and \( 3x + y^2 - 30 \):

1. \( 17 - x = (3x - y^2 - 2) - 17 \)
2. \( (3x - y^2 - 2) - 17 = (3x + y^2 - 30) - (3x - y^2 - 2) \)

Let's simplify these:

#### Equation 1:
\[
17 - x = 3x - y^2 - 19
\]
\[
36 = 4x - y^2 \quad \text{(Equation A)}
\]

#### Equation 2:
\[
(3x - y^2 - 2) - 17 = (3x + y^2 - 30) - (3x - y^2 - 2)
\]
Simplifying:
\[
3x - y^2 - 19 = 2y^2 - 28
\]
\[
y^2 = x + 3 \quad \text{(Equation B)}
\]

### Step 3: Substitute Equation B into Equation A
\[
36 = 4x - (x + 3)
\]
\[
36 = 3x - 3
\]
\[
39 = 3x
\]
\[
x = 13
\]

### Step 4: Find \( y^2 \)
Substitute \( x = 13 \) into Equation B:
\[
y^2 = 13 + 3 = 16
\]
\[
y = 4 \quad \text{(since \( y^2 = 16 \))}
\]

### Step 5: Find the terms of the sequence and their sum
The four terms are:
\[
x = 13, \quad 17, \quad 3x - y^2 - 2 = 39 - 16 - 2 = 21, \quad 3x + y^2 - 30 = 39 + 16 - 30 = 25
\]

So, the four terms are \( 13 \), \( 17 \), \( 21 \), and \( 25 \).

The sum is:
\[
13 + 17 + 21 + 25 = 76
\]

### Step 6: Determine divisibility
\( 76 \) is divisible by \( 2 \) and \( 4 \).

Thus, **the sum of the four numbers is divisible by 4**.

---

Would you like further details or have any questions? Here are some follow-up questions:

1. What is an arithmetic sequence, and how is it different from a geometric sequence?
2. How would the solution change if the terms were in a decreasing arithmetic sequence?
3. What other properties can be derived from the arithmetic sequence in this problem?
4. How can we verify the solution to ensure that the values satisfy all conditions?
5. Could we solve this problem using another approach, such as substitution from different sequences?

**Tip:** Always verify your results by substituting the values back into the original conditions to ensure they hold true.

Explore an arithmetic sequence problem where four consecutive terms are given, and their sum's divisibility is determined. Step-by-step solution provided with detailed calculations. Suitable for high school level students.

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