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We are given a sequence \( a_n = (x, y, z-1, y+6, 3x, \dots) \), and it is stated that this sequence is an arithmetic progression (AP). The goal is to find the value of \( x^2 + y^2 + z^2 \).

### Step 1: Understanding the Problem
For an arithmetic progression (AP), the difference between consecutive terms must remain constant. In other words, the difference between \( a_{n+1} \) and \( a_n \) is constant for all terms in the sequence.

The sequence is given as:
\[ a_n = (x, y, z-1, y+6, 3x, \dots) \]

Let’s break this down term by term:
1. First term: \( a_1 = x \)
2. Second term: \( a_2 = y \)
3. Third term: \( a_3 = z - 1 \)
4. Fourth term: \( a_4 = y + 6 \)
5. Fifth term: \( a_5 = 3x \)

### Step 2: Set Up the Arithmetic Progression Condition
For the sequence to be an arithmetic progression, the common difference \( d \) should be the same between consecutive terms. Hence, we can set up the following equations based on the AP condition:

1. \( y - x = d \)
2. \( (z - 1) - y = d \)
3. \( (y + 6) - (z - 1) = d \)
4. \( 3x - (y + 6) = d \)

### Step 3: Solve the System of Equations
Let’s solve these equations step by step.

From the first equation:
\[ y - x = d \]
\[ d = y - x \tag{1} \]

From the second equation:
\[ (z - 1) - y = d \]
\[ d = z - 1 - y \tag{2} \]

From the third equation:
\[ (y + 6) - (z - 1) = d \]
\[ d = y + 6 - z + 1 \]
\[ d = y + 7 - z \tag{3} \]

From the fourth equation:
\[ 3x - (y + 6) = d \]
\[ d = 3x - y - 6 \tag{4} \]

Now, we can equate the different expressions for \( d \) from the equations (1), (2), (3), and (4).

### Step 4: Equating and Solving for \( x \), \( y \), and \( z \)

From equations (1) and (2):
\[ y - x = z - 1 - y \]
\[ 2y = z - 1 + x \]
\[ z = 2y + 1 - x \tag{5} \]

From equations (1) and (3):
\[ y - x = y + 7 - z \]
\[ -x = 7 - z \]
\[ z = x + 7 \tag{6} \]

Now, using equation (5) and substituting \( z = x + 7 \):
\[ x + 7 = 2y + 1 - x \]
\[ 2x = 2y - 6 \]
\[ x = y - 3 \tag{7} \]

Substitute \( x = y - 3 \) into equation (6):
\[ z = (y - 3) + 7 \]
\[ z = y + 4 \tag{8} \]

### Step 5: Calculate \( x^2 + y^2 + z^2 \)

Now that we have \( x = y - 3 \), \( z = y + 4 \), we can compute \( x^2 + y^2 + z^2 \):

\[
x^2 + y^2 + z^2 = (y - 3)^2 + y^2 + (y + 4)^2
\]
\[
= (y^2 - 6y + 9) + y^2 + (y^2 + 8y + 16)
\]
\[
= y^2 - 6y + 9 + y^2 + y^2 + 8y + 16
\]
\[
= 3y^2 + 2y + 25
\]

Thus, the value of \( x^2 + y^2 + z^2 \) is:
\[
\boxed{3y^2 + 2y + 25}
\]

Would you like further details on any step, or do you have other questions?

### 5 Related Questions:
1. How can you identify if a sequence is an arithmetic progression (AP)?
2. Can this approach be extended to geometric progressions?
3. What is the difference between solving linear systems and quadratic systems of equations?
4. How would the problem change if the sequence was a geometric progression instead of arithmetic?
5. How can we check if our solution to \( x^2 + y^2 + z^2 \) is correct for specific values of \( x \), \( y \), and \( z \)?

### Tip:
When solving systems of equations derived from arithmetic progressions, always simplify equations step-by-step to avoid errors and check for consistent solutions across different equations.

Si la siguiente sucesión es una progresión aritmética, halle el valor de x^2 + y^2 + z^2. (a_n) = (x, y, z-1, y+6, 3x, ...)

In this problem, we solve an arithmetic progression involving variables x, y, and z. Through systems of equations, we determine the value of x^2 + y^2 + z^2. Key concepts include solving linear systems and using arithmetic progression formulas.

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