If x, y, z are in arithmetic progression and a is the arithmetic mean of x and y and b is the arithmetic mean of y and z, then prove that y is the arithmetic mean of a and b.

Options

A. `x=y lt z`
B. `x= y gt z`
C. `x lt z lt y`
D. `x gt z lt y`

Explanation

To prove that \( y \) is the arithmetic mean of \( a \) and \( b \), we will follow these steps:

### Step 1: Understand the Arithmetic Progression
Since \( x, y, z \) are in arithmetic progression, we can express this relationship mathematically. The definition of arithmetic progression states that the middle term \( y \) is the average of the other two terms \( x \) and \( z \).

\[ y = \frac{x + z}{2} \]

### Step 2: Define the Arithmetic Means
Next, we need to find the arithmetic means \( a \) and \( b \):

- \( a \) is the arithmetic mean of \( x \) and \( y \):
\[ a = \frac{x + y}{2} \]

- \( b \) is the arithmetic mean of \( y \) and \( z \):
\[ b = \frac{y + z}{2} \]

### Step 3: Add \( a \) and \( b \)
Now, we will add \( a \) and \( b \):

\[ a + b = \frac{x + y}{2} + \frac{y + z}{2} \]

Since the denominators are the same, we can combine the fractions:

\[ a + b = \frac{x + y + y + z}{2} = \frac{x + 2y + z}{2} \]

### Step 4: Substitute \( x + z \)
From our earlier step, we know that \( x + z = 2y \). We can substitute this into our equation:

\[ a + b = \frac{(2y) + 2y}{2} = \frac{4y}{2} = 2y \]

### Step 5: Find the Arithmetic Mean of \( a \) and \( b \)
To find the arithmetic mean of \( a \) and \( b \), we take:

\[ \text{Arithmetic Mean of } a \text{ and } b = \frac{a + b}{2} \]

Substituting \( a + b = 2y \):

\[ \text{Arithmetic Mean of } a \text{ and } b = \frac{2y}{2} = y \]

### Conclusion
Thus, we have shown that \( y \) is indeed the arithmetic mean of \( a \) and \( b \):

\[ y = \frac{a + b}{2} \]

This completes the proof.

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