Final answer:
To find the next number in this non-linear sequence, we need to identify the pattern or rule that governs the sequence. Upon analyzing the given sequence, we can observe that the first difference between consecutive terms is increasing by 5 each time. By constructing a quadratic equation and solving for the next term, we find that the next number in the sequence is 72.
Step-by-step explanation:
To find the next number in this non-linear sequence, we need to identify the pattern or rule that governs the sequence. Upon analyzing the given sequence, we can observe that the first difference between consecutive terms is increasing by 5 each time: 8 - 3 = 5, 19 - 8 = 11, 25 - 19 = 6, 33 - 25 = 8. The second difference between these first differences is constant, which means the sequence is described by a quadratic equation.
Considering this, we can construct a quadratic equation using the first differences. Let's call the position of the term 'n'. Our first differences are: 5, 11, 6, 8. The equation can be written as f(n) = an^2 + bn + c. We substitute the values of 'n' and the first differences into the equation to solve for 'a', 'b', and 'c'. Once we have the equation, we can find the next term in the sequence by substituting 'n' with 6 and evaluating f(n).
After finding the quadratic equation to be f(n) = 2n^2 - 3n + 6, we can substitute 'n' with 6 to find the next term. Plugging in 6 into the equation gives us f(6) = 2(6)^2 - 3(6) + 6 = 72.