Number Patterns with Mixed Operations and Number Formats

The first move with any number pattern is to write the gaps between the terms. Three, ten, seventeen, twenty-four: the gap is seven every time, so the rule is add seven and the next term is thirty-one. Gaps that are all the same mean adding or subtracting. Gaps that grow — four, twelve, thirty-six — mean multiplying, and here each term is three times the last. Write every gap, not just the first two; one, two, four, eight is doubling, but one, two, four, seven has gaps of one, two, three and is something else. Then test your rule on every term before you use it.

When the gaps double, the rule is double-then-something. Five, eleven, twenty-three, forty-seven: the gaps are six, twelve, twenty-four. Five doubled is ten and eleven is one more; eleven doubled is twenty-two and twenty-three is one more. Double, then add one — next term ninety-five. Six, eight, twelve, twenty, thirty-six has gaps two, four, eight, sixteen: double, then take four. Multiply a term by whatever the gaps multiply by, see what is left to add or subtract, and check the rule on a second pair before you trust it.

Decimals and fractions follow the same rules; the only extra step is making the gaps visible. Zero point two five, zero point five, zero point seven five, one: line the points up and each gap is zero point two five, so the next term is one point two five. A half, three quarters, one, one and a quarter: rename to quarters — two, three, four, five quarters — and the rule is add a quarter, so the next is one and a half. Zero point two, zero point four, zero point eight, one point six is doubling. A pattern does not stop at a whole number; it carries straight through.

For an adding pattern you can reach a far term without listing every one. Start at three, add seven, tenth term: from the first term to the tenth is nine jumps — fence posts and gaps — so the tenth term is three plus nine sevens, sixty-six. The fiftieth is three plus forty-nine sevens. Written as a formula, the nth term is seven n minus four, and checking it on the second term, fourteen minus four is ten, confirms it. The difference between the fourth and sixth terms is two jumps, fourteen. Doubling patterns have no shortcut like this at this level; list them.

A test question asks whether a number belongs to a pattern. Six, ten, fourteen, eighteen: the nth term is four n plus two. Fifty: four n plus two equals fifty gives n equals twelve, a whole number, so yes, fifty is the twelfth term. Thirty-two gives n equals seven and a half, so no. The quicker way is remainders: every term is two more than a multiple of four, so subtract two and see whether four divides what is left. And sometimes a property settles it without any formula — in three, five, seven every term is odd, so twenty-four is never there. Quiz and exercises now.