Final answer:
To find the sum of the first 28 terms of an arithmetic progression given the 14th term and the sum of the first 13 terms, we use the formulas for the nth term and the sum of n terms of an A.P. By solving simultaneously for the first term and the common difference, we can calculate the sum of the first 28 terms which is 882.
Step-by-step explanation:
The question is asking to find the sum of the first 28 terms of an arithmetic progression (A.P) given that the 14th term is 55 and the sum of the first 13 terms is 351. We can use the formula for the nth term of an A.P which is Tn = a + (n - 1)d, where a is the first term and d is the common difference. We can also use the formula for the sum of the first n terms of an A.P, which is Sn = n/2 (2a + (n - 1)d).
Since the 14th term is 55, we can write the equation for the 14th term as 55 = a + 13d. Since the sum of the first 13 terms is 351, we can put this in the sum formula to get 351 = 13/2 (2a + 12d). We can solve these two equations simultaneously to find a and d, and then use the sum formula to find the sum of the first 28 terms.
After finding a and d, we substitute them into the sum formula for n = 28 to calculate the sum of the first 28 terms. The correct answer to the question is 882, which is option a.