Build tile patterns, spot the jump and write the rule, like 3n + 2. Find missing terms, test whether a pattern is linear and beat the clock.
Start here. A sequence is a list of numbers that follows a rule. Each number is called a term. The first term is term 1, the next is term 2, and so on. In 3, 8, 13, 18 the terms go up by the same jump every time.
The common difference. The jump between neighbours is the common difference. In 3, 8, 13, 18 it is 5. A sequence with the same jump every time is called linear (or arithmetic). To find the next term, add the jump to the last term.
Tile patterns. In the workshop the blue tiles grow: every step adds one more blue column. The orange tiles stay the same. If each column has 3 blue tiles and there are 2 orange tiles, term 1 has 3 + 2 = 5 tiles, term 2 has 6 + 2 = 8 tiles and term 3 has 9 + 2 = 11 tiles.
The nth-term rule. A rule such as 3n + 2 tells you any term without counting. The letter n is the term number. 3n means 3 × n. For term 10 you work out 3 × 10 + 2 = 32. The number in front of n is the jump. The number added at the end is what you get if you go one step back before term 1.
Is a number in the sequence? Take the rule 4n + 2 and the number 50. Undo the + 2 to get 48, then divide by 4 to get 12. Term 12 is 50. If the division does not come out as a whole number, the number is not in the sequence.
Not linear. Square numbers 1, 4, 9, 16 have jumps of 3, 5, 7, so the jump keeps changing. The rule is n². Triangle numbers 1, 3, 6, 10 and doubling patterns 2, 4, 8, 16 are not linear either. In a Fibonacci-style sequence each term is the sum of the two terms before it.
The 20th term. You do not need to list 20 terms. Term 20 is 19 jumps after term 1, so add 19 times the jump to the first term.
A sequence is a list of numbers in a fixed order that follows a pattern or rule, such as 2, 4, 6, 8.
The nth term is a rule that gives any term from its position n. For the sequence 5, 8, 11, 14 the nth-term rule is 3n + 2.
3n means 3 × n. In algebra we leave out the multiplication sign. For n = 4, 3n = 12.
Find the common difference. That is the number in front of n. Then work out what is left to make term 1 correct. For 7, 12, 17 the jump is 5, so the rule starts 5n. 5 × 1 = 5 but term 1 is 7, so add 2: the rule is 5n + 2.
2n − 1 gives 1, 3, 5, 7 and so on. The even numbers come from 2n.
Just starting: next numbers, the jump and tile patterns. I know the basics: use, find and write nth-term rules. Challenge me: linear or not, squares, doubling, Fibonacci and the 20th term.
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