To solve the problem step by step, we need to analyze the conditions given for the sequences and derive the relationships between the variables \( a, x, b, y, z \).
### Step 1: Establish Relationships from A.P., G.P., and H.P.
1. **A.P. Condition**: Since \( a, x, b \) are in A.P., we have:
\[
2x = a + b \quad \text{(1)}
\]
2. **G.P. Condition**: Since \( a, y, b \) are in G.P., we have:
\[
y^2 = ab \quad \text{(2)}
\]
3. **H.P. Condition**: Since \( a, z, b \) are in H.P., we have:
\[
z = \frac{2ab}{a + b} \quad \text{(3)}
\]
### Step 2: Substitute the Given Conditions
We are given two additional conditions:
- \( x = y + 2 \quad \text{(4)} \)
- \( a = 5z \quad \text{(5)} \)
### Step 3: Express \( z \) in terms of \( a \) and \( b \)
From equation (3):
\[
z = \frac{2ab}{a + b}
\]
Substituting \( z \) into equation (5):
\[
a = 5 \left( \frac{2ab}{a + b} \right)
\]
This simplifies to:
\[
a(a + b) = 10ab
\]
Rearranging gives:
\[
a^2 + ab - 10ab = 0 \quad \Rightarrow \quad a^2 - 9ab = 0
\]
Factoring out \( a \):
\[
a(a - 9b) = 0
\]
Since \( a \neq 0 \), we have:
\[
a = 9b \quad \text{(6)}
\]
### Step 4: Substitute \( a \) in Terms of \( b \) into Equation (1)
Substituting equation (6) into equation (1):
\[
2x = 9b + b = 10b \quad \Rightarrow \quad x = 5b \quad \text{(7)}
\]
### Step 5: Substitute \( a \) into Equation (2)
Substituting equation (6) into equation (2):
\[
y^2 = ab = 9b^2 \quad \Rightarrow \quad y = 3b \quad \text{(8)}
\]
### Step 6: Substitute \( y \) into Equation (4)
Using equation (4):
\[
x = y + 2 \quad \Rightarrow \quad 5b = 3b + 2
\]
Solving for \( b \):
\[
5b - 3b = 2 \quad \Rightarrow \quad 2b = 2 \quad \Rightarrow \quad b = 1
\]
### Step 7: Find \( a, x, y, z \)
Using \( b = 1 \) in equation (6):
\[
a = 9b = 9 \cdot 1 = 9
\]
Using \( b = 1 \) in equations (7) and (8):
\[
x = 5b = 5 \cdot 1 = 5
\]
\[
y = 3b = 3 \cdot 1 = 3
\]
Now, substitute \( a \) and \( b \) into equation (3) to find \( z \):
\[
z = \frac{2ab}{a + b} = \frac{2 \cdot 9 \cdot 1}{9 + 1} = \frac{18}{10} = 1.8
\]
### Summary of Values
- \( a = 9 \)
- \( b = 1 \)
- \( x = 5 \)
- \( y = 3 \)
- \( z = 1.8 \)
### Final Result
The relationships are:
\[
x > y > z \quad \Rightarrow \quad 5 > 3 > 1.8
\]
