Final answer:
The explicit formula for the given sequence is Sn = 18 + 1/2n(13 + (n - 1)), which accounts for the increasing differences in the original sequence starting from 18.
Step-by-step explanation:
The student asked for an explicit formula for the sequence 18, 25, 33, 42, 52, 63, 75. This sequence represents the differences between consecutive elements increasing by one each time, starting with 7 (25 - 18), then 8 (33 - 25), etc.
We can derive an explicit formula by noticing that these differences form an arithmetic sequence. The nth term of the sequence can be obtained using the formula for the sum of the first n terms of an arithmetic sequence, which is:
- Find the nth difference by starting with 7 and adding n-1 each time.
- Sum these differences and add it to the first term, which is 18, to find the nth term of the sequence.
The explicit formula is, therefore, the sum of an arithmetic series starting with 7 and with a common difference of 1, plus the first term (18). This can be represented by the formula Sn = 18 + ½n(13 + (n - 1)), as the arithmetic series starts at 7 (the first difference) and at the nth term is at (7 + (n - 1)).