Final answer:
The sequence 2, 14, 36, 68 follows a quadratic pattern and its nth term is given by the formula T(n) = 2n² - n + 1. To find a specific term, such as the nth term, we substitute the value of n into the formula. For example, for n=5, the nth term would be 46.
Step-by-step explanation:
The sequence given is 2, 14, 36, 68, which seems to follow a quadratic pattern. To find the expression for the nth term of this sequence, we need to establish a rule that fits the given terms of the sequence. Observing the sequence, we can see that the differences between consecutive terms are increasing linearly: 12, 22, 32, ...—suggesting a quadratic pattern. Since the sequence does not start at 0, we should account for the constant term when deriving our nth term formula.
Let's attempt to derive the formula:
For n=1, we have 2; for n=2, we have 14, and so on.
So the nth term, T(n), could be represented in the form of an² + bn + c.
Using the sequence numbers for n=1, 2, 3, and 4, and solving the system of equations, we find that a=2, b=-1, and c=1.
Thus, the nth term of the sequence is T(n) = 2n² - n + 1.
To find a specific term in the sequence, such as the nth term, we simply substitute the value of n into the nth term formula. For example, for n=5, T(5) = 2(5)² - 5 + 1 = 2(25) - 5 + 1 = 50 - 5 + 1 = 46.