If three numbers are in G.P., then the numbers obtained by adding the middle number to each of these numbers are in

Options

A. A.P.
B. G.P.
C. H.P.
D. none of these

Answer

To solve the problem, we need to determine if the numbers obtained by adding the middle number of three numbers in a geometric progression (G.P.) to each of those numbers are in an arithmetic progression (A.P.), geometric progression (G.P.), or harmonic progression (H.P.).

### Step-by-Step Solution:

1. **Define the three numbers in G.P.**:
Let the three numbers in G.P. be \( \frac{a}{r}, a, ar \), where \( a \) is the middle term and \( r \) is the common ratio.

2. **Add the middle number to each of the three numbers**:
We add the middle number \( a \) to each of the three numbers:
- First number: \( \frac{a}{r} + a = \frac{a + ar}{r} = \frac{a(1 + r)}{r} \)
- Second number: \( a + a = 2a \)
- Third number: \( ar + a = a(r + 1) \)

Thus, the new numbers are:
\[ \frac{a(1 + r)}{r}, \quad 2a, \quad a(r + 1) \]

3. **Let’s denote the new numbers**:
Let:
- \( A = \frac{a(1 + r)}{r} \)
- \( B = 2a \)
- \( C = a(r + 1) \)

4. **Check if these numbers are in H.P.**:
To check if \( A, B, C \) are in H.P., we need to verify if:
\[ \frac{1}{A} + \frac{1}{C} = \frac{2}{B} \]

Calculating \( \frac{1}{A} \) and \( \frac{1}{C} \):
\[ \frac{1}{A} = \frac{r}{a(1 + r)}, \quad \frac{1}{C} = \frac{1}{a(r + 1)} \]

Now adding these:
\[ \frac{1}{A} + \frac{1}{C} = \frac{r}{a(1 + r)} + \frac{1}{a(r + 1)} = \frac{r + 1}{a(1 + r)} \]

Now calculate \( \frac{2}{B} \):
\[ \frac{2}{B} = \frac{2}{2a} = \frac{1}{a} \]

5. **Set the equation**:
Now we need to check if:
\[ \frac{r + 1}{a(1 + r)} = \frac{1}{a} \]

Cross-multiplying gives:
\[ (r + 1) = (1)(1 + r) \]

This simplifies to:
\[ r + 1 = 1 + r \]
which is true.

6. **Conclusion**:
Since the condition for H.P. is satisfied, the numbers obtained by adding the middle number to each of the three numbers in G.P. are in H.P.

### Final Answer:
The numbers obtained by adding the middle number to each of the three numbers in G.P. are in **Harmonic Progression (H.P.)**.