Final answer:
The nth term rule for the quadratic sequence -6, -3, 2, 9, 18 is T(n) = 2.5n² - 9.5n + 2. This was determined by finding the pattern of differences between terms, which suggested a relationship with square numbers and then solving a system of equations.
Step-by-step explanation:
To find the nth term rule for the quadratic sequence -6, -3, 2, 9, 18, first we must determine the pattern of the sequence. We can see that the differences between each term are increasing by a constant amount, which is characteristic of a quadratic sequence. Specifically, the differences of the terms are 3, 5, 7, and 9, which are consecutive odd numbers. This suggests that the sequence is based on the sums of odd numbers, which we can relate to square numbers.
The sum of the first n odd numbers is n². For this sequence, each term is the sum of odd numbers with an initial offset. To get the quadratic term, we look at the pattern of square numbers. Starting with 1, the square numbers increase by the next odd number: 1, 4 (1+3), 9 (4+5), 16 (9+7), and so on. This pattern reflects the relationship that the nth term is n² plus an adjustment.
To find the specific quadratic rule for our sequence, we can create an equation for the nth term, T(n), which will include the squared term plus a linear adjustment:
T(n) = an² + bn + c. To find the coefficients a, b, and c, we plug in the first few terms of the sequence. For instance, using the first term:
- -6 = a(1)² + b(1) + c
- -3 = a(2)² + b(2) + c
- 2 = a(3)² + b(3) + c
By solving the system of equations, we find that a = 2.5, b = -9.5, and c = 2. Therefore, the nth term rule for this sequence is:
T(n) = 2.5n² - 9.5n + 2