Let `a_(1),a_(2),a_(3),a_(4),a_(5)` be a five term geometric sequence satisfies the condition `0 lt a_(1) lt a_(2) lt a_(3) lt a_(4) lt 100` , where each term is an integar . Then the number of five terms in geometric progression are

Options

A. 6
B. 7
C. 8
D. 10

Answer

To solve the problem, we need to find the number of five-term geometric sequences \( a_1, a_2, a_3, a_4, a_5 \) that satisfy the conditions \( 0 < a_1 < a_2 < a_3 < a_4 < 100 \) where each term is an integer.

### Step-by-Step Solution:

1. **Define the Terms of the Geometric Sequence**:
Since \( a_1, a_2, a_3, a_4, a_5 \) are in geometric progression, we can express them in terms of the middle term \( a_3 \) and the common ratio \( r \):
\[ a_1 = \frac{a_3}{r^2}, \quad a_2 = \frac{a_3}{r}, \quad a_3 = a_3, \quad a_4 = ar, \quad a_5 = ar^2 \]

2. **Establish the Conditions**:
From the conditions \( 0 < a_1 < a_2 < a_3 < a_4 < 100 \):
- Since \( a_1 = \frac{a_3}{r^2} > 0 \), \( a_3 > 0 \) and \( r > 1 \).
- The last term \( a_5 = ar^2 < 100 \).

3. **Express the Conditions**:
- From \( a_5 < 100 \):
\[ a_3 r^2 < 100 \]
- From \( a_1 > 0 \):
\[ a_3 > 0 \]

4. **Determine the Values of \( r \)**:
Since \( r \) must be greater than 1 and an integer, the possible values of \( r \) can be 2, 3, 4, etc. We will check these values to find valid \( a_3 \).

5. **Calculate the Maximum \( a_3 \)**:
For each integer \( r \):
- \( r = 2 \):
\[ a_3 \cdot 2^2 < 100 \implies a_3 < 25 \quad (a_3 = 1, 2, \ldots, 24) \]
- \( r = 3 \):
\[ a_3 \cdot 3^2 < 100 \implies a_3 < \frac{100}{9} \approx 11.11 \quad (a_3 = 1, 2, \ldots, 11) \]
- \( r = 4 \):
\[ a_3 \cdot 4^2 < 100 \implies a_3 < 6.25 \quad (a_3 = 1, 2, 3, 4, 5, 6) \]
- \( r = 5 \):
\[ a_3 \cdot 5^2 < 100 \implies a_3 < 4 \quad (a_3 = 1, 2, 3) \]
- \( r = 6 \):
\[ a_3 \cdot 6^2 < 100 \implies a_3 < \frac{100}{36} \approx 2.78 \quad (a_3 = 1, 2) \]
- \( r = 7 \) and higher will yield \( a_3 < 2 \) which is not valid since \( a_3 \) must be a positive integer.

6. **Count the Valid Sequences**:
Now we count the valid \( a_3 \) values for each \( r \):
- For \( r = 2 \): 24 values
- For \( r = 3 \): 11 values
- For \( r = 4 \): 6 values
- For \( r = 5 \): 3 values
- For \( r = 6 \): 2 values

Adding these gives:
\[ 24 + 11 + 6 + 3 + 2 = 46 \]

### Final Answer:
The total number of five-term geometric sequences that satisfy the given conditions is **46**.