The recursive formula for an arithmetic sequence has two parts: a starting value and a rule that builds each new term from the one before it. In general notation, it looks like this:
a(1) = A, a(n) = a(n − 1) + d
Here, A is the first term of the sequence, and d is the common difference, the constant amount added (or subtracted) each time. Both pieces are required. Without the starting value, the rule alone could describe infinitely many different sequences.
How the Formula Works
Take the sequence 3, 5, 7, 9, 11 … The difference between every pair of consecutive terms is 2, so d = 2. The recursive formula is:
a(1) = 3, a(n) = a(n − 1) + 2
To find the fifth term, you work forward one step at a time. Start with a(1) = 3. Then a(2) = 3 + 2 = 5, a(3) = 5 + 2 = 7, a(4) = 7 + 2 = 9, and a(5) = 9 + 2 = 11. Each term depends entirely on the term right before it, which is what makes the formula “recursive.”
Finding the Common Difference
The common difference is the value you get when you subtract any term from the term that follows it: d = a(n + 1) − a(n). If every pair of consecutive terms gives you the same result, the sequence is arithmetic and that result is your d.
The common difference can be positive, negative, or zero. In the sequence 100, 96, 92, 88 …, each term is 4 less than the previous one, so d = −4. The recursive formula would be a(1) = 100, a(n) = a(n − 1) − 4. A sequence like −2, −5, −8, −11 … has d = −3. When d is negative, the sequence decreases. When d is zero, every term is the same number.
How to Write One From a List of Numbers
If you’re given a sequence and need to write its recursive formula, there are only two things to figure out:
- Identify the first term. This becomes a(1) in your formula.
- Calculate the common difference. Subtract any term from the next one. Check at least two pairs to confirm the difference is constant.
For example, given the sequence 10, 3, −4, −11 …, the first term is 10. The difference between consecutive terms is 3 − 10 = −7, and −4 − 3 = −7, confirming d = −7. The recursive formula: a(1) = 10, a(n) = a(n − 1) − 7.
Why the First Term Matters
The rule a(n) = a(n − 1) + 2 by itself is incomplete. It tells you to “add 2 to the previous term,” but it never says what the previous term started as. The sequences 3, 5, 7, 9 … and 100, 102, 104, 106 … both add 2 each time, yet they’re completely different sequences. Defining a(1) pins the formula to one specific sequence. Without it, you can’t compute a single term.
Recursive vs. Explicit Formulas
An arithmetic sequence can also be described by an explicit formula: a(n) = a(1) + d(n − 1). The explicit version lets you jump directly to any term. If you need the 100th term, you plug in n = 100 and get your answer in one calculation.
The recursive formula, by contrast, requires you to calculate every term that comes before the one you want. To find the 100th term recursively, you’d need to compute all 99 preceding terms first. That makes it inefficient for large values of n.
So why use recursive formulas at all? They’re often more intuitive because they describe the pattern directly: “start here, then keep adding this amount.” They’re useful when you’re building out a sequence step by step, listing the first 10 or 15 terms. They also translate naturally into spreadsheets or simple programs where each row references the row above it. The explicit formula is better when you need a single term far down the line without generating everything before it.
Converting Between the Two Forms
Going from recursive to explicit is straightforward. If your recursive formula is a(1) = 3, a(n) = a(n − 1) + 2, then the first term is 3 and the common difference is 2. Plug those into the explicit template: a(n) = 3 + 2(n − 1), which simplifies to a(n) = 2n + 1.
Going the other direction, pull the first term and the common difference out of the explicit formula. If a(n) = 5n − 8, find a(1) by plugging in n = 1: a(1) = 5(1) − 8 = −3. The coefficient on n is the common difference, so d = 5. The recursive version: a(1) = −3, a(n) = a(n − 1) + 5.