Final answer:
The value of S25 for the given arithmetic sequence [6, 13, 20, 27,......] is 2250, calculated using the sum formula for an arithmetic sequence.
Step-by-step explanation:
The given sequence is an arithmetic sequence because there is a constant difference between successive terms. To determine the value of S25, which is the sum of the first 25 terms, we can use the formula for the sum of the first n terms of an arithmetic sequence: Sn = n/2 (a1 + an), where a1 is the first term, an is the last term, and n is the number of terms.
To find a25, we use the term formula for an arithmetic sequence: an = a1 + (n-1)d, where d is the common difference. For the given sequence [6, 13, 20, 27], d = 13 - 6 = 7. Therefore, a25 = 6 + (25-1)*7 = 6 + 24*7 = 6 + 168 = 174.
Now, using the sum formula: S25 = 25/2 * (6 + 174) = 25/2 * 180 = 25 * 90 = 2250. The value of S25 for the sequence is 2250.