Final answer:
To find the sum of the first 16 terms of the A.P., we solve two equations derived from the sum of the first four and eight terms. With the first term and the common difference found, we use the sum formula for an A.P. and determine that the sum of the first 16 terms is 608. The correct answer is (c) 608.
Step-by-step explanation:
The student has asked to find the sum of the first 16 terms of an arithmetic progression (A.P.) given that the sum of the first four terms is 56, and the sum of the first eight terms is 176. We can use the formula for the sum of the first n terms of an A.P., which is Sn = n/2 * (2a + (n-1)d), where n is the number of terms, a is the first term, and d is the common difference.
Let's first establish two equations based on the given information:
- S4 = 56 = 4/2 * (2a + (4-1)d)
- S8 = 176 = 8/2 * (2a + (8-1)d)
Solving these two equations simultaneously gives us the values of a and d. Once we have the first term and the common difference, we can then find the sum of the first 16 terms using the formula S16 = 16/2 * (2a + (16-1)d).
After calculating the correct values for a and d, and using the formula, we find that the sum of the first 16 terms of this A.P. is 608, which is option (c).