The difference between the nth term and (n-1)th term of a sequence is independent of n. then the sequence follows which one of the following ?

Options

A. A) AP
B. B) GP
C. C) HP
D. D) None of these

Answer

To solve the problem, we need to analyze the given condition that the difference between the nth term and the (n-1)th term of a sequence is independent of n. Let's denote the nth term of the sequence as \( a_n \).

### Step-by-Step Solution:

1. **Understanding the Difference**:
We start by expressing the difference between the nth term and the (n-1)th term:
\[ d_n = a_n - a_{n-1} \]
According to the problem, this difference \( d_n \) is independent of \( n \). This means that for all values of \( n \), \( d_n \) is a constant value, say \( k \).

2. **Setting the Constant Difference**:
Since \( d_n \) is constant, we can write:
\[ d_n = k \quad \text{for all } n \]
This implies:
\[ a_n - a_{n-1} = k \]

3. **Finding a General Formula for the Sequence**:
We can express the terms of the sequence in terms of the first term \( a_1 \):
- For \( n = 2 \):
\[ a_2 = a_1 + k \]
- For \( n = 3 \):
\[ a_3 = a_2 + k = (a_1 + k) + k = a_1 + 2k \]
- For \( n = 4 \):
\[ a_4 = a_3 + k = (a_1 + 2k) + k = a_1 + 3k \]
- Continuing this pattern, we can generalize:
\[ a_n = a_1 + (n-1)k \]

4. **Identifying the Type of Sequence**:
The expression \( a_n = a_1 + (n-1)k \) is the general form of an arithmetic progression (AP), where:
- \( a_1 \) is the first term,
- \( k \) is the common difference.

5. **Conclusion**:
Since the difference between consecutive terms is constant, we conclude that the sequence follows an Arithmetic Progression (AP).

### Final Answer:
The sequence follows **Option A: AP (Arithmetic Progression)**.