Growing Patterns: Table, Rule, Far Term

A growing pattern is a picture that gets bigger by a rule. Squares in a row made of matchsticks: one square is four sticks, two squares are seven, three are ten, because each new square shares a side with the last and costs only three. Draw the next one or two, count them carefully, and put the counts in a table against the pattern number. The table shows the gap: plus three every time. Triangles in a row go up by two; the tiles around the edge of a square go up by four.

The gap in the table is the multiplier in the rule. Sticks go up by three, so the rule begins three times n. Three times one is three, but the first term is four, so add one: three n plus one. Check it on every row — three times five plus one is sixteen. In words, three times the number of squares plus one, and you can even see the plus one: it is the first stick on the left, before any square adds its three. Triangles, gap two and first term three, give two n plus one. Border tiles, gap four and first term eight, give four n plus four.

Once you have the rule, a far term is just the rule with the number put in. Fifty squares: three times fifty plus one, one hundred and fifty-one sticks. The tenth pattern has thirty-one, the twentieth sixty-one, the hundredth three hundred and one. The tenth square of border tiles has four times ten plus four, forty-four. Before you trust the rule for a big number, check it on a small one you can draw; and remember that a rule found from two rows of a table can be wrong — three checked rows is safe.

The reverse question gives the count and asks which pattern it is. A hundred sticks: three n plus one equals a hundred, so three n is ninety-nine and n is thirty-three. Undo the rule as in the balance lesson: take away the one, divide by three. If n comes out as a whole number, the count is in the pattern; if it is a fraction, the count falls between two patterns — no border square uses exactly fifty tiles, because four n plus four equals fifty gives eleven and a half.

Tiles around the edge of a square: three by three has eight, four by four has twelve, five by five has sixteen. Gap four, first term eight, rule four n plus four, where n is the pattern number. You can see the rule in the picture: four sides of n tiles, plus the four corners that belong to no side. Seeing why the rule is what it is lets you write it without a table, and it catches a miscount, because a rule that does not match the picture is wrong somewhere. Quiz and exercises now.