Generating Sequences

In an arithmetic sequence, the hop from each number to the next is always the same size. That hop is called the common difference. Look at 3, 5, 7, 9. Every hop is plus 2, so the common difference is 2. You can keep the sequence going by adding 2 again and again. You can also jump straight to any spot. If a sequence starts at 3 and adds 2 each time, the number in position n is 2n + 1. Try it: position 1 gives 2 x 1 + 1 = 3, and position 4 gives 2 x 4 + 1 = 9.

In a geometric sequence, you multiply by the same number to get each next term. That number is called the common ratio. In 1, 2, 4, 8, every step is times 2, so the common ratio is 2. Arithmetic sequences grow by steady hops, but geometric sequences grow faster and faster, because each jump is a multiplication, not an addition. To check if a sequence is geometric, divide neighbors: 2 / 1 = 2, 4 / 2 = 2, 8 / 4 = 2. Same ratio every time means geometric. If the divisions do not match, the sequence is something else, like the square numbers 1, 4, 9, 16.

What is the difference between an arithmetic sequence and a geometric one? In an arithmetic sequence you add the same number each time to get the next term. In a geometric sequence you multiply by the same number each time. How do you build an arithmetic sequence? Pick any first term, then pick a common difference, the number you will keep adding. Start at 4 and choose 5, and you get 4, 9, 14, 19. How do you build a geometric sequence? Pick a first term and a common ratio to keep multiplying by. Start at 4 with ratio 3 and you get 4, 12, 36, 108. And some sequences are neither type, like 1, 1, 2, 3, 5, where the rule changes from step to step.

A recursive formula, which is another name for a term-to-term rule, gives you two things: the first term, and the rule for getting each term from the one before it. Here is a recursive formula for the sequence 3, 5, 7: the first term is 3, and you add 2 to get the next term. To find the 10th term you can count up one by one, making nine hops of plus 2, and you land on 21. Recursive formulas are great for watching a pattern grow step by step. They are also how computers often store sequences, because each new term only needs the term right before it.

An arithmetic sequence is a list of numbers where each one is the last one plus a fixed amount. That fixed amount is the common difference. Say a plant grows 2 cm every week, starting at 5 cm tall. Its heights form an arithmetic sequence: 5, 7, 9, 11, and so on, with common difference 2. There are two ways to describe the same sequence. The recursive way says: start at 5, add 2 each time. The explicit way uses the position: the height in week n is 2n + 3. Both ways give the same list, so you can pick the one that fits the problem you are solving.