The explicit formula for the given sequence can be derived by analyzing the pattern of the numbers. The sequence presented is 2, 9, 28, 65, 126, and can be noted as the series of values determined by the formula a(n) = n^3 – n + 2, where n is the term number starting from 1.
Here’s how it works for each term:
- For n = 1: a(1) = 1^3 – 1 + 2 = 2
- For n = 2: a(2) = 2^3 – 2 + 2 = 9
- For n = 3: a(3) = 3^3 – 3 + 2 = 28
- For n = 4: a(4) = 4^3 – 4 + 2 = 65
- For n = 5: a(5) = 5^3 – 5 + 2 = 126
This clearly shows that each term in the sequence corresponds with the formula, confirming its validity. The formula captures the growth rate of the values in the sequence, showcasing how they increase cubically as n progresses. Therefore, the explicit formula generating the infinite sequence is indeed a(n) = n^3 – n + 2.