Given:
- The first, second, and fourth terms of the A.P. are in G.P.
- The fifth term of the A.P. is 13.

To find:
- The sum of the first 10 terms of the A.P.

Approach:
1. Let the first term of the A.P be a and the common difference be d.
2. Then, the second term is a + d, and the fourth term is a + 3d.
3. Since the first, second, and fourth terms are in G.P., we can write the equation as (a + d)^2 = (a)(a + 3d).
4. Expanding and simplifying this equation, we get a^2 + 2ad + d^2 = a^2 + 3ad.
5. Simplifying further, we get 2ad + d^2 = 3ad.
6. Rearranging the terms, we have d^2 = ad.
7. Since the common difference cannot be zero, we can divide both sides of the equation by d to get d = a.
8. Substituting the value of d in the equation d^2 = ad, we get a^2 = a^2.
9. This implies that a can have any value.

Let's assume a = 1 for simplicity.

Calculation:
1. Given that the fifth term of the A.P. is 13, we can write the equation a + 4d = 13.
2. Substituting the value of a = 1, we get 1 + 4d = 13.
3. Solving this equation, we find d = 3.
4. The first ten terms of the A.P. are: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28.
5. The sum of the first ten terms can be calculated using the formula for the sum of an A.P.: Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the first n terms, a is the first term, and d is the common difference.
6. Substituting the values, we get S10 = (10/2)(2(1) + (10-1)3) = 5(2 + 27) = 5(29) = 145.

Therefore, the sum of the first 10 terms of the A.P. is 145, which is not one of the given options. However, since we assumed a = 1, the answer may vary depending on the value of a. Thus, none of the given options is correct.