An A.P. and a G.P. with positive terms have the same number of terms and their first terms as well as last terms are equal. Then what can be said about the sum of the A.P. and G.P. If `A_s ` and `G_s` denote the sum of the A.P. and G.P. respectively :

Options

A. `A_S ge G_s`
B. `A_s le G_s`
C. `A_s = G_s`
D. can not be determined

Answer

To solve the problem, we need to analyze the properties of an Arithmetic Progression (A.P.) and a Geometric Progression (G.P.) that have the same number of terms, with equal first and last terms. We will denote the first term of both sequences as \( a \) and the last term as \( l \).

### Step-by-step Solution:

1. **Define the A.P.**:
- Let the first term of the A.P. be \( a \) and the last term also be \( a \).
- Let the number of terms in the A.P. be \( n \).
- The common difference of the A.P. can be denoted as \( d \).
- The terms of the A.P. can be expressed as:
\[ a, \, a + d, \, a + 2d, \, \ldots, \, a + (n-1)d \]
- Since the last term is also \( a \), we have:
\[ a + (n-1)d = a \implies (n-1)d = 0 \]
- Since \( d \) must be zero for positive terms, this means all terms in the A.P. are equal to \( a \).

2. **Calculate the Sum of the A.P.**:
- The sum \( A_s \) of the A.P. is given by:
\[ A_s = \frac{n}{2} \times (\text{first term} + \text{last term}) = \frac{n}{2} \times (a + a) = \frac{n}{2} \times 2a = n \cdot a \]

3. **Define the G.P.**:
- Let the first term of the G.P. also be \( a \) and the last term also be \( a \).
- The common ratio of the G.P. can be denoted as \( r \).
- The terms of the G.P. can be expressed as:
\[ a, \, ar, \, ar^2, \, \ldots, \, ar^{n-1} \]
- Since the last term is also \( a \), we have:
\[ ar^{n-1} = a \implies r^{n-1} = 1 \]
- Since \( r \) must be positive, this implies \( r = 1 \).

4. **Calculate the Sum of the G.P.**:
- The sum \( G_s \) of the G.P. is given by:
\[ G_s = \frac{a(r^n - 1)}{r - 1} \]
- Since \( r = 1 \), we can calculate the sum as follows:
\[ G_s = na \quad (\text{since all terms are equal to } a) \]

5. **Compare the Sums**:
- We find that:
\[ A_s = n \cdot a \quad \text{and} \quad G_s = n \cdot a \]
- Therefore, we conclude that:
\[ A_s = G_s \]

### Conclusion:
The sum of the A.P. and the sum of the G.P. are equal when both sequences have the same number of terms, and their first and last terms are equal.