If `x ,y ,z` are in A.P. and `A_1` is the A.M. of x and y and `A_2` is the A.M. of y and z , then prove that the A.M. of `A_1 `and `A_2` is y .

Explanation

To prove that the arithmetic mean of \( A_1 \) and \( A_2 \) is \( y \), given that \( x, y, z \) are in arithmetic progression (A.P.), we can follow these steps:

### Step 1: Define the Arithmetic Means
Given that \( x, y, z \) are in A.P., we know:
- The arithmetic mean \( A_1 \) of \( x \) and \( y \) is given by:
\[ A_1 = \frac{x + y}{2} \]
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