Let `a_(1),a_(2),a_(3), …, a_(10)` be in G.P. with `a_(i) gt 0` for i=1, 2, …, 10 and S be te set of pairs (r, k), r, k `in` N (the set of natural numbers)
for which `|(log_(e)a_(1)^(r)a_(2)^(k),log_(e)a_(2)^(r)a_(3)^(k),log_(e)a_(3)^(r)a_(4)^(k)),(log_(e)a_(4)^(r)a_(5)^(k),log_(e)a_(5)^(r)a_(6)^(k),log_(e)a_(6)^(r)a_(7)^(k)),(log_(e)a_(7)^(r)a_(8)^(k),log_(e)a_(8)^(r)a_(9)^(k),log_(e)a_(9)^(r)a_(10)^(k))|` = 0. Then the number of elements in S is

Options

A. Infinitely many
B. 4
C. 10
D. 2

Answer

To solve the problem, we need to analyze the determinant given in the question and find the conditions under which it equals zero. The sequence \( a_1, a_2, \ldots, a_{10} \) is in geometric progression (G.P.) with \( a_i > 0 \) for \( i = 1, 2, \ldots, 10 \).

### Step-by-Step Solution:

1. **Understanding the G.P.**:
Since \( a_1, a_2, \ldots, a_{10} \) are in G.P., we can express them as:
\[ a_i = a_1 r^{i-1} \quad \text{for } i = 1, 2, \ldots, 10 \]
where \( r \) is the common ratio and \( a_1 > 0 \).

2. **Writing the Determinant**:
The determinant we need to analyze is:
\[ D = \begin{vmatrix} \log_e(a_1^r a_2^k) & \log_e(a_2^r a_3^k) & \log_e(a_3^r a_4^k) \\ \log_e(a_4^r a_5^k) & \log_e(a_5^r a_6^k) & \log_e(a_6^r a_7^k) \\ \log_e(a_7^r a_8^k) & \log_e(a_8^r a_9^k) & \log_e(a_9^r a_{10}^k) \end{vmatrix} \]

3. **Applying Logarithmic Properties**:
Using the property of logarithms, we can rewrite each term:
\[ \log_e(a_i^r a_{i+1}^k) = r \log_e(a_i) + k \log_e(a_{i+1}) \]
Thus, we can express the determinant in terms of logarithms of the \( a_i \).

4. **Column Operations**:
Perform column operations to simplify the determinant:
- Let \( C_2 = C_2 - C_1 \)
- Let \( C_3 = C_3 - C_2 \)

After these operations, we will find that the determinant can be expressed in terms of differences of logarithms.

5. **Identifying Linear Dependence**:
After performing the column operations, we will find that two columns become identical or linearly dependent. This is because the logarithmic terms will have a common structure due to the G.P. property.

6. **Conclusion**:
Since the determinant becomes zero when any two columns are identical, we conclude that the determinant \( D = 0 \) for all pairs \( (r, k) \) in \( \mathbb{N} \).

7. **Counting the Elements in Set \( S \)**:
Since \( r \) and \( k \) can take any natural number value, the number of pairs \( (r, k) \) is infinite. Thus, the number of elements in the set \( S \) is infinite.

### Final Answer:
The number of elements in \( S \) is infinite.