3.1 Consider the quadratic number pattern: \( 3 ; 7 ; 12 ; \ldots \)...

Did you know that the given quadratic number pattern can be broken down by examining the differences between consecutive numbers? The first differences are \(4\) (from \(3\) to \(7\)) and \(5\) (from \(7\) to \(12\)). If you calculate the second differences, which is simply the difference between the first differences, you’ll find they are constant at \(1\). This tells you it's a quadratic sequence, specifically of the form \(an^2 + bn + c\)! Now, let's apply this to predict the next term! To find the next number in the pattern after \(12\), continue the first differences: the next difference would be \(6\) (adding the second difference of \(1\) to the last first difference of \(5\)). So, \(12 + 6\) gives you \(18\). The sequence now extends to \(3; 7; 12; 18\) – and you can keep going from there!