Final answer:
The sum of the first n terms of an A.P. can be calculated using the standard formula involving the number of terms n, the first term a, and the common difference d. Given the sum of the first four and first fourteen terms, we can derive a system of equations to find the values of a and d and subsequently express the sum of the first n terms.
Step-by-step explanation:
To find the sum of the first n terms of an arithmetic progression (A.P.), we use the formula for the sum of an A.P., which is Sn = n/2 [2a + (n-1)d], where Sn is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms. Given that the sum of the first four terms is 40 (S4 = 40), and the sum of the first 14 terms is 280 (S14 = 280), we can set up two equations to solve for a and d:
1. S4 = 4/2 [2a + (4-1)d] = 40
2. S14 = 14/2 [2a + (14-1)d] = 280
Then, using these values, we can express the sum of the first n terms as Sn = n/2 [2a + (n-1)d].