Final answer:
Camila concluded the sum of the first n positive odd integers is n^2, most likely through empirical observation, by noticing that the sequential sum of odd numbers forms perfect squares, as seen in a pattern or table created during her investigation.
Step-by-step explanation:
Camila concluded that the sum of the first n positive odd integers is n^2, which indicates that she must have observed a specific pattern or sequence in her investigation. This conclusion comes about from recognizing that summing odd numbers sequentially (1, 3, 5, 7, ...) forms perfect squares (1 = 1^2, 1+3 = 4 = 2^2, 1+3+5 = 9 = 3^2, and so on). To obtain this result, one method Camila could have used is mathematical induction, a powerful technique for proving the truth of infinitely many statements. However, the question implies that Camila used empirical observation to arrive at her conclusion.
Empirical observation involves looking at specific instances and noticing a pattern or trend. It is possible that Camila also constructed a table that showed the sums of odd integers and noted that each sum was a square number, which is consistent with empirical observation. Therefore, it is likely that she looked at the actual outcomes and observed that the sums matched n^2 for the first few positive integers.