Writing Recursive Formulas from a Sequence
A recursive formula defines each term of a sequence by referencing one or more previous terms. That's it. You need two things: a starting value and a rule that tells you how to get from term n-1 to term n. Nothing fancy. Most Algebra 2 students I've worked with struggle not with the concept itself but with setting it up correctly under time pressure, so let me walk through how I actually approach this when someone hands me a problem. Here's the format you'll use. For a sequence like 3, 7, 11, 15... the recursive formula is f(1) = 3 and f(n) = f(n-1) + 4 for n greater than 1. You write both parts. Leaving out the base case is the most common error I see, and it's an easy point loss on tests. The base case anchors the whole thing. Without it, you have a rule but no way to start calculating. I worked with a student last month who had a sequence: 2, 6, 18, 54. She wrote f(n) = 3 * f(n-1) with f(1) = 2. That was technically correct but she lost points because the problem asked for the formula in terms of n explicitly and she needed to verify her index matched. Small thing, but it comes up. Always double-check what the question is actually asking for before you turn it in.
For arithmetic sequences, the recursion is straightforward addition or subtraction. If the common difference is d, then f(n) = f(n-1) + d. For geometric sequences, you multiply by the common ratio r, so f(n) = r * f(n-1). This distinction matters because students often mix them up when the problem gives you a list of numbers instead of telling you upfront which type it is. Here's something that trips people up: some sequences aren't purely arithmetic or geometric. Take 1, 1, 2, 3, 5, 8. That's Fibonacci. The rule is f(n) = f(n-1) + f(n-2). You're referencing two previous terms, not one. This is perfectly valid in Algebra 2, and you'll see it on exams. Just make sure you state both base cases: f(1) = 1 and f(2) = 1. Missing one base case for a two-term recursion is an instant deduction. Let me give you a concrete example from scratch. Say you're given the sequence: 5, 9, 13, 17. First, find the pattern. Each term goes up by 4. So the common difference is 4. Base case: f(1) = 5. Recursive rule: f(n) = f(n-1) + 4. Done. Write it out fully: f(1) = 5, f(n) = f(n-1) + 4 for n 2.
Now a slightly harder one. Sequence: 3, -1, -5, -9. Common difference is -4. f(1) = 3, f(n) = f(n-1) - 4 for n 2. Same process, just a negative difference. Nothing changes in the method.
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When Recursive Formulas Fall Apart
I need to be honest about where this approach gets ugly. Recursive formulas are fine for finding the next few terms, but if you need to find, say, the 50th term, computing every single term from 1 to 50 is painful and unnecessary. That's where the closed-form formula comes in. For the arithmetic sequence above, the closed form is f(n) = 4n + 1. Plug in n = 50 and you get 201 immediately. No stepping through 49 intermediate terms. Another edge case I ran into recently: a sequence where the common difference itself changes. Like 2, 5, 10, 17, 26. The differences are 3, 5, 7, 9. That's not constant, so it's not arithmetic. The recursive formula would need to reference the pattern in the differences, which gets complicated fast. In my experience, these problems usually want you to find a quadratic closed form instead, like f(n) = n² + 1. Testing this: f(1) = 2, f(2) = 5, f(3) = 10. It works. The recursive approach becomes messy here, and the closed form is cleaner. If your problem gives you f(3) and f(4) instead of f(1), you can still write the recursive formula but you need to adjust your base case accordingly. Some textbooks will accept f(3) = 5, f(n) = f(n-1) + 4 for n 4, but others require the base case to start at n = 1. Know what your teacher or textbook expects. This inconsistency causes real confusion and costs points for no good reason.
Another thing: recursive formulas don't always have a nice closed form. Some sequences, like certain combinatorial ones, are much easier to compute recursively than any closed formula could capture. Don't force a closed form when it doesn't exist or when it's impractical. The recursive definition might actually be the simplest correct answer.
Quick Reference for Common Types
Arithmetic sequence with first term a and common difference d: f(1) = a, f(n) = f(n-1) + d. Geometric sequence with first term a and common ratio r: f(1) = a, f(n) = r * f(n-1). Fibonacci-type with starting values a and b: f(1) = a, f(2) = b, f(n) = f(n-1) + f(n-2).
These are the three types you'll encounter in Algebra 2. Master those and you've covered 95 percent of what shows up on tests. Anything beyond that is usually a bonus problem or a competition-level question.
