Consider the quadratic sequence shown in the grid. 11, nine, six, two. Find the next term.
First, we need to remember what a quadratic sequence is. It’s a sequence where the difference between each term will change by a constant amount. To go from 11 to nine, you subtract two. Moving from nine to six, we subtract three. Moving from six to two, we subtract four. And now, we begin to see the quadratic sequence. Each time, we’re subtracting one more than the previous term. Negative three is one less than negative two. Negative four is one less than negative three.
Following this pattern, we should subtract five from two. Since we took four away from six, we’ll need to take five away from two. In a quadratic sequence, we call this amount the second difference, the constant amount that is changing each time. Back to our question, we need to find this fifth term. Two minus five equals negative three. If we consider this subtraction two minus five on a number line, we’ll start at two. And subtraction is represented by moving left on a number line.
Five places to the left of two on the number line is negative three, which confirms two minus five is negative three. And negative three is the next term in this [quadratic] sequence.