To solve the problem, we need to analyze the given information and derive the necessary conclusions step by step.
### Step-by-Step Solution:
1. **Understanding the Given Information**:
- We know that a sequence is an Arithmetic Progression (A.P.) if the difference between consecutive terms is constant.
- The nth term of an A.P. can be expressed as:
\[
t_n = a + (n-1)d
\]
- The sum of the first n terms of an A.P. is given by:
\[
S_n = \frac{n}{2} \left(2a + (n-1)d\right) = \frac{n}{2} \left(a + l\right)
\]
where \( l \) is the last term.
2. **Understanding Geometric Progression (G.P.)**:
- If \( a_1, a_2, \ldots, a_n \) are in G.P., then the ratio of consecutive terms is constant.
- This means:
\[
\frac{a_2}{a_1} = \frac{a_3}{a_2} = \ldots = \frac{a_n}{a_{n-1}} = r \quad (\text{common ratio})
\]
3. **Logarithmic Transformation**:
- We take logarithms of each term in the G.P.:
\[
\log(a_1), \log(a_2), \log(a_3), \ldots, \log(a_n)
\]
- Since the logarithm of a product is the sum of logarithms, we can express the logarithmic terms in terms of the common ratio \( r \).
4. **Establishing the Relationship**:
- The logarithmic terms can be rewritten as:
\[
\log(a_1), \log(a_1) + \log(r), \log(a_1) + 2\log(r), \ldots, \log(a_1) + (n-1)\log(r)
\]
- This shows that the logarithmic terms form an A.P. with the first term \( \log(a_1) \) and common difference \( \log(r) \).
5. **Reciprocal of Logarithmic Terms**:
- Now, if we take the reciprocals of these logarithmic terms:
\[
\frac{1}{\log(a_1)}, \frac{1}{\log(a_2)}, \frac{1}{\log(a_3)}, \ldots, \frac{1}{\log(a_n)}
\]
- The reciprocals of terms in an A.P. form a Harmonic Progression (H.P.).
6. **Conclusion**:
- Since the logarithmic terms \( \log(a_1), \log(a_2), \ldots, \log(a_n) \) are in A.P., their reciprocals \( \frac{1}{\log(a_1)}, \frac{1}{\log(a_2)}, \ldots, \frac{1}{\log(a_n)} \) are in H.P.
### Final Answer:
The logarithmic terms \( \log(a_1), \log(a_2), \ldots, \log(a_n) \) are in A.P., and their reciprocals are in H.P.
