Choose the correct alternative. The sum of the first 20 terms and the first 50 terms of an A.P. is 670 and 3925 respectively. Find the sum of first five terms of a G.P. whose first term is same as that of the A.P. and whose common ratio is equal to the common difference of the A.P.

To find the sum of the first five terms of the G.P., we first need the common difference (d) of the A.P. and the first term (a). Given the equations for the sum of the A.P.: 1. \( S_{20} = \frac{20}{2} \cdot (2a + 19d) = 670 \) So, \( 20a + 190d = 670 \) — (1) 2. \( S_{50} = \frac{50}{2} \cdot (2a + 49d) = 3925 \) So, \( 50a + 1225d = 3925 \) — (2) Now we solve these equations simultaneously. From equation (1), we can simplify: \( 20a + 190d = 670 \) → \( 2a + 19d = 67 \) And from equation (2): \( 50a + 1225d = 3925 \) → \( 10a + 245d = 785 \) Next, we can multiply equation (1) by 5 to align it with equation (2): \( 10a + 95d = 335 \) — (3) Now subtract equation (3) from equation (2): \( (10a + 245d) - (10a + 95d) = 785 - 335 \) Thus, \( 150d = 450 \) \( d = 3 \) Now, substitute \( d \) back into equation (1) to find \( a \): \( 20a + 190(3) = 670 \) \( 20a + 570 = 670 \) \( 20a = 100 \) \( a = 5 \) Now for the G.P. with first term \( a = 5 \) and common ratio \( r = d = 3 \): The sum of the first 5 terms of the G.P. can be calculated using the formula: \[ S_n = a \frac{(r^n - 1)}{r - 1} \] Thus, \[ S_5 = 5 \frac{(3^5 - 1)}{3 - 1} = 5 \frac{(243 - 1)}{2} = 5 \cdot \frac{242}{2} = 5 \cdot 121 = 605 \] Therefore, the sum of the first five terms of the G.P. is \( 605 \).