Final answer:
The correct notation for the sum of n terms in a series is ∑ k=1 to n ak, which is option A. This represents adding up all terms ak from the first term when k=1 to the nth term. Other options include summation from 0 or to infinity, which do not match the sum of a finite number of n terms.
Step-by-step explanation:
The student is asking about the notation for the sum of a series. The correct notation for the sum, Sn, where n represents the number of terms, is option A: ∑ k=1 to n ak. This notation indicates that you are summing the sequence of terms ak from k=1 up to and including n. Options B, C, and D contain summation indices that start from 0 or go to infinity, which are not specific to the sum of n terms.
For instance, in a series of odd numbers (1, 3, 5, ...), if we sum n terms, we can express the summation as ∑ k=1 to n (2k-1). The claim in the box that this leads to n² is a reflection of the pattern that the sum of the first n odd numbers is always a perfect square. The student's statement that 2[n + n + ... + n + n] = 2n² further illustrates the property that the sum of n terms of natural numbers is n(n+1)/2, which is a different concept but related in providing an example of closed-form expressions for certain series.