The number 1,5 and 25 can be three terms (not necessarily consecutive) of

Options

A. only one AP
B. more than one but finite terms of APs
C. infinite number of APs
D. Finite number of GPs

Answer

To determine whether the numbers 1, 5, and 25 can be three terms (not necessarily consecutive) of a sequence, we will check if they can form a geometric progression (GP).

### Step-by-Step Solution:

1. **Identify the Terms**: The three numbers given are 1, 5, and 25.

2. **Recall the Condition for GP**: For three numbers \( a, b, c \) to be in a GP, they must satisfy the condition:
\[ b^2 = a \cdot c \]
where \( a \) is the first term, \( b \) is the second term, and \( c \) is the third term.

3. **Assign Values**: Assign the values:
- \( a = 1 \)
- \( b = 5 \)
- \( c = 25 \)

4. **Calculate \( b^2 \)**: Calculate \( b^2 \):
\[ b^2 = 5^2 = 25 \]

5. **Calculate \( a \cdot c \)**: Calculate \( a \cdot c \):
\[ a \cdot c = 1 \cdot 25 = 25 \]

6. **Check the Condition**: Now check if \( b^2 = a \cdot c \):
\[ 25 = 25 \]
The condition is satisfied.

7. **Conclusion**: Since the condition \( b^2 = a \cdot c \) holds true, the numbers 1, 5, and 25 can indeed be terms of a geometric progression.

### Final Answer:
The numbers 1, 5, and 25 can be three terms of a geometric progression (GP).

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