First three of four numbers are in A.P., the last three in H.P. The four numbers are proportional.

Answer

To solve the problem, we need to establish the relationships between the four numbers based on the given conditions: the first three numbers are in Arithmetic Progression (A.P.), the last three numbers are in Harmonic Progression (H.P.), and the four numbers are proportional.

Let's denote the four numbers as \( A, B, C, D \).

### Step 1: Express the numbers in A.P.
Since \( A, B, C \) are in A.P., we can express them as:
- \( B = A + d \)
- \( C = A + 2d \)

where \( d \) is the common difference of the A.P.

### Step 2: Express the numbers in H.P.
Since \( B, C, D \) are in H.P., we can use the property of H.P. that states the reciprocals of the numbers are in A.P. Therefore, we can write:
- The reciprocals: \( \frac{1}{B}, \frac{1}{C}, \frac{1}{D} \) are in A.P.

This means:
\[ \frac{1}{C} - \frac{1}{B} = \frac{1}{D} - \frac{1}{C} \]

### Step 3: Substitute for \( B \) and \( C \)
Substituting \( B \) and \( C \) into the H.P. condition:
\[ \frac{1}{A + 2d} - \frac{1}{A + d} = \frac{1}{D} - \frac{1}{A + 2d} \]

### Step 4: Solve for \( D \)
Cross-multiplying gives:
\[ \frac{(A + d) - (A + 2d)}{(A + d)(A + 2d)} = \frac{(A + 2d) - D}{(A + 2d)D} \]
This simplifies to:
\[ \frac{-d}{(A + d)(A + 2d)} = \frac{(A + 2d) - D}{(A + 2d)D} \]

### Step 5: Cross-multiply and simplify
Cross-multiplying leads to:
\[ -d \cdot (A + 2d)D = (A + d)(A + 2d) \cdot ((A + 2d) - D) \]

### Step 6: Establish proportionality
Since the four numbers are proportional, we can express this as:
\[ \frac{A}{B} = \frac{C}{D} \]
Substituting the values we have:
\[ \frac{A}{A + d} = \frac{A + 2d}{D} \]

### Step 7: Solve for \( D \)
Cross-multiplying gives:
\[ A \cdot D = (A + d)(A + 2d) \]
Expanding and simplifying will yield the value of \( D \).

### Conclusion
We have established the relationships and derived the necessary equations to prove that the four numbers are proportional.