Final answer:
The next five terms of the given sequence are 13, 25, 39, 55, and 73. The nth term formula is related to the arithmetic pattern that emerges from the differences between consecutive terms of the sequence.
Step-by-step explanation:
The student has provided a sequence of numbers and is asking two questions. First, they want to know the next five terms of the sequence −17, −15, −11, −5, 3, …. Second, they are looking for the formula, based on the partial sum, for the nth term of the sequence.
By observing the pattern in the given sequence, we can see that with each term, the number being added increases by 2 more than the last increment. This suggests that the differences between consecutive terms follow an arithmetic pattern:
- From −17 to −15, we add 2.
- From −15 to −11, we add 4.
- From −11 to −5, we add 6.
- From −5 to 3, we add 8.
The next difference is expected to be 10, followed by 12, 14, 16, and 18 respectively for the next five terms in the sequence. Applying these differences to the last known term (3), we get the following next five terms:
- 3 + 10 = 13
- 13 + 12 = 25
- 25 + 14 = 39
- 39 + 16 = 55
- 55 + 18 = 73
The nth term of the sequence can be described using the formula for the nth partial sum of an arithmetic series, where each term increases by an increment that is itself increasing by 2. The partial sum is equal to half the product of the number of terms and the sum of the first and last term. To find the nth term specifically, we would examine the sequence of differences to deduce a formula based on n terms and the respective arithmetic pattern.