Final answer:
The nᵗʰ term rule for the given quadratic sequence 6, 12, 22, 36, 54, ... is T = 4n + 8.
Step-by-step explanation:
The given sequence is 6, 12, 22, 36, 54, ...
We observe that the difference between consecutive terms is increasing by 6 each time. The first term has a difference of 6, the second term has a difference of 10 (6 + 4), the third term has a difference of 14 (10 + 4), and so on. This indicates that the sequence is a quadratic sequence.
To find the nᵗʰ term rule for this quadratic sequence, we need to find a formula that relates the term number (n) to the term value (T).
Let's consider the differences between consecutive terms: 6, 10, 14, ...
The differences themselves form a linear sequence with a common difference of 4. We can determine the nth term rule for this linear sequence using the formula:
T = a + (n - 1)d
where T is the nth term, a is the first term, n is the term number, and d is the common difference.
Using this formula, we can find the nth term rule for the differences:
T = 6 + (n - 1)4
Simplifying this equation gives us:
T = 4n + 2
Now, we can find the nth term rule for the original sequence by adding the nth term rule for the differences to the nth term rule for the original sequence:
T = 6 + (4n + 2)
Simplifying this equation gives us:
T = 4n + 8
Therefore, the nᵗʰ term rule for this quadratic sequence is T = 4n + 8.