A sequence is called an A.P. if the difference of a term and the previous term is always same i.e. if `a_(n+1)-a_(n)` = constant (common difference) for all `n in N`.
For an A.P. whose first term is 'a' and common difference is 'd' has its `n^("th")` term as `t_(n)=a+(n-1)d` Sum of n terms of an A.P. whose first is a, last term is I and common difference is d is
`S_(n)=(n)/(2)(2a+(n-1)d)`
`=(n)/(2)(a+a+(n-1)d)=(n)/(2)(a+l)`.
`S_(r)` denotes the sum of first r terms of a G.P., then `S_(n),S_(2n)-S_(n),S_(3n)-S_(2n)` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. none of these

Answer

To solve the problem, we need to analyze the sums of the first \( n \) terms of a geometric progression (G.P.) and compare them to determine the nature of the sequence formed by \( S_n, S_{2n} - S_n, S_{3n} - S_{2n} \).

### Step-by-Step Solution:

1. **Understand the Sum of G.P.:**
The sum of the first \( n \) terms of a geometric progression (G.P.) is given by:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
where \( a \) is the first term and \( r \) is the common ratio.

2. **Calculate \( S_{2n} \):**
Using the formula for the sum of a G.P., we find \( S_{2n} \):
\[ S_{2n} = \frac{a(1 - r^{2n})}{1 - r} \]

3. **Calculate \( S_{3n} \):**
Similarly, we calculate \( S_{3n} \):
\[ S_{3n} = \frac{a(1 - r^{3n})}{1 - r} \]

4. **Find \( S_{2n} - S_n \):**
Now, we compute \( S_{2n} - S_n \):
\[ S_{2n} - S_n = \frac{a(1 - r^{2n})}{1 - r} - \frac{a(1 - r^n)}{1 - r} \]
Simplifying this gives:
\[ S_{2n} - S_n = \frac{a(r^n - r^{2n})}{1 - r} = \frac{ar^n(1 - r^n)}{1 - r} \]

5. **Find \( S_{3n} - S_{2n} \):**
Next, we compute \( S_{3n} - S_{2n} \):
\[ S_{3n} - S_{2n} = \frac{a(1 - r^{3n})}{1 - r} - \frac{a(1 - r^{2n})}{1 - r} \]
Simplifying this gives:
\[ S_{3n} - S_{2n} = \frac{a(r^{2n} - r^{3n})}{1 - r} = \frac{ar^{2n}(1 - r^n)}{1 - r} \]

6. **Establish the Relationship:**
Now we have:
- \( S_n = \frac{a(1 - r^n)}{1 - r} \)
- \( S_{2n} - S_n = \frac{ar^n(1 - r^n)}{1 - r} \)
- \( S_{3n} - S_{2n} = \frac{ar^{2n}(1 - r^n)}{1 - r} \)

Notice that:
\[ S_{2n} - S_n = r^n S_n \]
\[ S_{3n} - S_{2n} = r^{2n} S_n \]

7. **Identify the Sequence:**
The expressions \( S_n, S_{2n} - S_n, S_{3n} - S_{2n} \) can be written as:
- First term: \( S_n \)
- Second term: \( r^n S_n \)
- Third term: \( r^{2n} S_n \)

This indicates that the sequence \( S_n, S_{2n} - S_n, S_{3n} - S_{2n} \) is a geometric progression (G.P.) because each term is a constant multiple of the previous term.

### Conclusion:
Thus, the answer is that \( S_n, S_{2n} - S_n, S_{3n} - S_{2n} \) are in a G.P.