The sequence 0.3,0.33,0.333 ……., to ne terms is :

Options

A. an A.P
B. a G.P
C. an infinite G.P
D. None of these

Answer

To determine the nature of the sequence \(0.3, 0.33, 0.333, \ldots\) up to \(n\) terms, we will analyze whether it forms an Arithmetic Progression (AP), Geometric Progression (GP), or neither.

### Step-by-Step Solution:

1. **Identify the Terms of the Sequence:**
- The first term \(T_1 = 0.3\)
- The second term \(T_2 = 0.33\)
- The third term \(T_3 = 0.333\)
- Generally, the \(n\)-th term can be expressed as \(T_n = 0.\underbrace{33\ldots3}_{n \text{ threes}}\)

2. **Check for Arithmetic Progression (AP):**
- An AP has a constant difference between consecutive terms.
- Calculate the difference between the first two terms:
\[ T_2 - T_1 = 0.33 - 0.3 = 0.03 \]
- Calculate the difference between the second and third terms:
\[ T_3 - T_2 = 0.333 - 0.33 = 0.003 \]
- Since \(0.03 \neq 0.003\), the differences are not constant. Therefore, the sequence is not an AP.

3. **Check for Geometric Progression (GP):**
- A GP has a constant ratio between consecutive terms.
- Calculate the ratio of the second term to the first term:
\[ \frac{T_2}{T_1} = \frac{0.33}{0.3} = 1.1 \]
- Calculate the ratio of the third term to the second term:
\[ \frac{T_3}{T_2} = \frac{0.333}{0.33} \approx 1.0091 \]
- Since \(1.1 \neq 1.0091\), the ratios are not constant. Therefore, the sequence is not a GP.

4. **Conclusion:**
- Since the sequence does not satisfy the conditions for either an AP or a GP, we conclude that it is neither.

### Final Answer:
The sequence \(0.3, 0.33, 0.333, \ldots\) up to \(n\) terms is **none of these** (not an AP or GP).