If in any progressin the difference of any two consecutive terms bears a consant ratio to their product, then the given progression is in

Options

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

Answer

To solve the problem step-by-step, we will analyze the given condition and derive the conclusion about the type of progression.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given a sequence of terms \( a_1, a_2, a_3, \ldots, a_n \) and the condition states that the difference of any two consecutive terms bears a constant ratio to their product.

2. **Setting Up the Equation**:
Let's consider two consecutive terms \( a_n \) and \( a_{n-1} \). According to the problem, we can express this as:
\[ a_n - a_{n-1} = k \cdot (a_n \cdot a_{n-1}) \]
where \( k \) is some constant.

3. **Rearranging the Equation**:
Rearranging gives:
\[ a_n - a_{n-1} = k \cdot a_n \cdot a_{n-1} \]
This can be rewritten as:
\[ \frac{a_n - a_{n-1}}{a_n \cdot a_{n-1}} = k \]

4. **Considering Another Pair of Terms**:
Now, consider the next pair of terms, \( a_{n+1} \) and \( a_n \):
\[ a_{n+1} - a_n = k \cdot (a_{n+1} \cdot a_n) \]
Rearranging this gives:
\[ \frac{a_{n+1} - a_n}{a_{n+1} \cdot a_n} = k \]

5. **Setting Up the Ratio**:
Now, we can set up the ratio of the two equations:
\[ \frac{a_n - a_{n-1}}{a_{n+1} - a_n} = \frac{a_n \cdot a_{n-1}}{a_{n+1} \cdot a_n} \]
Simplifying this gives:
\[ \frac{a_n - a_{n-1}}{a_{n+1} - a_n} = \frac{a_{n-1}}{a_{n+1}} \]

6. **Cross-Multiplying**:
Cross-multiplying yields:
\[ (a_n - a_{n-1}) \cdot a_{n+1} = (a_{n+1} - a_n) \cdot a_{n-1} \]

7. **Identifying the Type of Progression**:
This equation indicates that the sequence follows a specific pattern. If we denote the terms in the sequence as \( a_1, a_2, a_3 \), we can generalize this to show that:
\[ \frac{1}{a_1} + \frac{1}{a_3} = \frac{2}{a_2} \]
This is the condition for three numbers \( a_1, a_2, a_3 \) to be in Harmonic Progression (HP).

8. **Conclusion**:
Therefore, we conclude that if the difference of any two consecutive terms bears a constant ratio to their product, then the given progression is in Harmonic Progression (HP).