Reversing a Rule: Each Term from the Previous

Some patterns are described by a rule that turns each term into the next. Start at six; each term is double the previous minus four. Six doubled is twelve, minus four is eight. Eight doubled is sixteen, minus four is twelve. Then twenty, then thirty-six, then sixty-eight. The two steps happen in order, double first and then subtract, and each term is built from the one before, so you write them all down as you go — the sixth term needs the fifth.

To go backwards, undo the rule: reverse the order of the steps and reverse each step. Double then take four becomes add four then halve. From thirty-six: add four, forty, halve, twenty — the term before. From twenty: twenty-four halved, twelve. Times three then plus one reverses to minus one then divide by three. The last thing the rule did is the first thing you undo; halving thirty-six first and then adding four gives twenty-two, which is wrong, and a fraction appearing mid-way is usually the sign that the order has slipped.

Given a later term, step back one term at a time until you reach the one you want. Rule double then take four, fifth term thirty-six: back to twenty, twelve, eight, six. Four steps from the fifth term to the first — count the gaps, not the terms. Then run six forwards as a check: eight, twelve, twenty, thirty-six. Write every term down; skipping one in your head is where these questions go wrong.

The difference between two terms is found by finding both terms and subtracting. Doubling from three, the fourth term is twenty-four and the sixth is ninety-six: seventy-two apart. For double then take four from six, the fourth is twenty, the sixth is sixty-eight, forty-eight apart. Adding rules have a shortcut — jumps times gap — but multiplying and two-step rules do not, so list. And to ask which term a number is, list until you reach it; if you overshoot, it is not in the sequence.

The rule comes in words, and the words fix the order. Each number is double the one before, minus four: double first. Each number is four less than double the one before: still double first. Each number is double the one before minus four, with the brackets around the subtraction: subtract first, then double — a different rule. When the question shows you the first few terms, test your reading of the rule on them before you go forwards or backwards. Quiz and exercises now.