The given sequence is 81, 108, 144, 192. To find a formula for this, we first look for a pattern in the sequence.
We start by observing the differences between the consecutive terms:
- 108 – 81 = 27
- 144 – 108 = 36
- 192 – 144 = 48
The differences between the terms are 27, 36, and 48. Next, we can look at the differences of these differences:
- 36 – 27 = 9
- 48 – 36 = 12
These secondary differences (9 and 12) suggest that the sequence may not be linear, so we check if it follows a polynomial pattern. The primary differences increase by 9 and 12, indicating a quadratic relationship.
To express this in a formula, we can assume a quadratic function of the form:
f(n) = an² + bn + c
By fitting the known values into this formula using n = 1, 2, 3, and 4 for terms 81, 108, 144, and 192 respectively, we can solve for a, b, and c.
Ultimately, we find that a functional representation that fits the sequence is:
f(n) = 27n² + 54n
Where n is the position in the sequence starting from 1. This formula successfully describes the relationship between the position and the values in the sequence.