Final answer:
The nth term of the quadratic sequence 3, 12, 27, 48 can be found by analyzing the second differences, which are constant indicating a quadratic pattern. By using the general form an^2 + bn + c and the values of the sequence, we can determine the coefficients a, b, and c, which allows us to express the nth term formula.
Step-by-step explanation:
To find the nth term of the quadratic sequence 3, 12, 27, 48, we first need to determine the pattern of the sequence. We look for the second differences because a quadratic sequence will have constant second differences. Let's calculate the differences:
- First differences: 12 - 3 = 9, 27 - 12 = 15, 48 - 27 = 21, ...
- Second differences: 15 - 9 = 6, 21 - 15 = 6, ...
Since the second differences are constant at 6, the sequence is indeed quadratic. The second difference, in this case, is 2 times 3, which indicates the coefficient of the n2 term is ½ × 6 = 3 (because the second difference is twice the a-n^2 coefficient).
The nth term of a quadratic sequence is generally given by an² + bn + c. To find a, b, and c, we can use the sequence and what we've already discovered:
- a = 3 (from the second difference)
- For n=1, the sequence gives the first term as 3 = a(1)² + b(1) + c = 3 + b + c. So b + c = 0.
- For n=2, the sequence gives the second term as 12 = a(2)² + b(2) + c = 12 + 2b + c. So 2b + c = 0.
Through this process, you can solve a system of equations to find specific values for b and c. After finding these, the nth term can be written down.