Final answer:
The nᵗʰ term of the quadratic sequence 9, 12, 17, 24, 33, ... is given by the rule 3n² + 2n + 4, which is derived by identifying the sequence as a quadratic sequence with a second difference of 2 and adjusting it to match the pattern of square numbers.
Step-by-step explanation:
The question asks for the nᵣ term rule of a quadratic sequence that begins with the numbers 9, 12, 17, 24, 33, .... To determine the rule, we first need to recognize the pattern within the sequence. A quadratic sequence is characterized by a second difference that is constant.
Here, the first differences are: 12 - 9 = 3, 17 - 12 = 5, 24 - 17 = 7, and 33 - 24 = 9. The second differences are: 5 - 3 = 2, 7 - 5 = 2, 9 - 7 = 2, confirming it is a quadratic sequence with a second difference of 2.
To find the nth term rule, it’s useful to relate this sequence to the sequence of square numbers, since we are dealing with a quadratic sequence.
Our sequence expansions and binomial theorem knowledge come into play here. After comparing with the sequence of square numbers (1, 4, 9, 16, ...) and adjusting for the pattern, we can determine that the nth term of the given quadratic sequence is 3n² + 2n + 4.