Final answer:
The nth term of the sequence 4, 5, 8, 13, 20 is given by a quadratic equation of the form an^2 + bn + c, which can be found using the second differences method. The nth term for this sequence is n^2 + 3n + 1.
Step-by-step explanation:
To find the nth term of the sequence 4, 5, 8, 13, 20, we need to determine the pattern that the sequence follows. By examining the sequence, we notice that the difference between each term is increasing by one as we move from one term to the next: 5 - 4 = 1, 8 - 5 = 3, 13 - 8 = 5, and so on. This suggests that the sequence is quadratic in nature. A quadratic sequence can generally be represented by an equation of the form an2 + bn + c.
The first step to finding this quadratic equation is to determine the second differences by taking the difference of consecutive first differences. If the second differences are constant, then the first term, a, is half the second difference. In this case, the second differences are all 2, which means a = 2 / 2 = 1. Hence, the term involving n2 is n2.
Next, we need to find the coefficients b and c. We can set up equations based on the known terms of the sequence and solve for b and c. After solving, we get that the quadratic equation for the nth term of this sequence is n2 + 3n + 1.