Recursive Formula Explained: Definition, Examples, and How to Use It

What is a recursive formula?

A recursive formula is a two-part definition of a sequence. The first part is the base case, the known starting value such as a_1. The second part is the recursive rule, an equation that expresses a general term a_n in terms of one or more earlier terms like a_(n-1) or a_(n-2).

The word recursive comes from the idea of a rule that refers back to itself. To find any term you must first know the term before it, which means you generally build the sequence one step at a time from the beginning. Without a base case the rule has nothing to start from, so the base case is not optional. A rule like a_n = a_(n-1) + 5 is satisfied by infinitely many different sequences (3, 8, 13... and 100, 105, 110... both obey it). The base case is what pins down one unique sequence.

• Base case: the starting term or terms, for example a_1 = 3.
• Recursive rule: how to build the next term, for example a_n = a_(n-1) + 5.
• Order matters: you find a_2 from a_1, then a_3 from a_2, and so on.

You will also meet the same idea written in function notation: f(n) = f(n-1) + 5 with f(1) = 3 means exactly the same thing as the subscript version. Some textbooks write a(n) instead of a_n, and some use u_n or t_n for the terms. The notation changes, the two-part structure never does.

The three classic recursive formulas

Most recursive sequences you study fall into three families. Each one keeps the same two-part shape but changes the rule that links one term to the next.

Arithmetic sequences

In an arithmetic sequence you add the same fixed amount, called the common difference d, every step. The recursive formula is a_n = a_(n-1) + d, with a_1 given. If a_1 = 2 and d = 4, the sequence runs 2, 6, 10, 14, and so on. A negative d simply counts down: with a_1 = 20 and d = -3 you get 20, 17, 14, 11. For a deeper look at this family, see our arithmetic sequence formula guide, or read Khan Academy's walkthrough of writing recursive formulas for arithmetic sequences.

Geometric sequences

In a geometric sequence you multiply by the same fixed factor, called the common ratio r, every step. The recursive formula is *a_n = a_(n-1) r**, with a_1 given. If a_1 = 3 and r = 2, the sequence runs 3, 6, 12, 24, and so on. A ratio between 0 and 1 shrinks the sequence instead: with a_1 = 80 and r = 0.5 you get 80, 40, 20, 10. This is the same pattern behind exponential growth and compound interest. The full breakdown lives in our geometric sequence formula guide.

The Fibonacci sequence

The Fibonacci sequence needs two earlier terms, not one. Its recursive formula is a_n = a_(n-1) + a_(n-2), with a_1 = a_2 = 1. Each term is the sum of the two before it, giving 1, 1, 2, 3, 5, 8, 13, 21, and so on. Because it depends on two predecessors, it needs two starting values rather than one.

Rules that are neither arithmetic nor geometric

Not every recursive rule adds or multiplies by a constant. The factorial is recursive too: n! = n (n-1)! with 0! = 1, so the multiplier changes at every step. Rules can also mix operations, such as a_n = 2 a_(n-1) + 3 with a_1 = 1, which gives 1, 5, 13, 29, 61. These sequences are still perfectly well defined, they just do not have the tidy explicit forms that arithmetic and geometric sequences do.

How do you use a recursive formula?

To use a recursive formula, start from the base case and apply the rule once per step until you reach the term you want. Each new term is computed from the term you just found, so you work strictly in order and never skip ahead.

Suppose you are given the arithmetic recursive formula a_n = a_(n-1) + 5 with a base case of a_1 = 3, and you want the first five terms.

1. Write down the base case. a_1 = 3.
2. Apply the rule for n = 2. a_2 = a_1 + 5 = 3 + 5 = 8.
3. Apply the rule for n = 3. a_3 = a_2 + 5 = 8 + 5 = 13.
4. Apply the rule for n = 4. a_4 = a_3 + 5 = 13 + 5 = 18.
5. Apply the rule for n = 5. a_5 = a_4 + 5 = 18 + 5 = 23.
6. Read off the sequence. The first five terms are 3, 8, 13, 18, 23.

Notice that to reach a_5 you had to pass through every term before it. That is the defining feature of recursion: there are no shortcuts, you climb the ladder one rung at a time. The same step by step style appears when you study an average rate of change across the points of a sequence.

A geometric example works the same way. Given a_n = a_(n-1) * 2 with a_1 = 5, you get a_2 = 10, a_3 = 20, a_4 = 40, and a_5 = 80. Each term is double the last. If you would rather generate dozens of terms without adding them by hand, drop the starting value and the rule into a calculator and read the whole sequence at once.

How do you write a recursive formula from a sequence?

To write a recursive formula from a list of terms, state the first term as the base case, then work out what operation turns each term into the next. Subtract consecutive terms to test for a common difference, and divide consecutive terms to test for a common ratio.

Take the sequence 7, 11, 15, 19, 23. Subtracting neighbours gives 11 - 7 = 4, 15 - 11 = 4, and 19 - 15 = 4. The difference is constant, so this is arithmetic with d = 4, and the recursive formula is a_n = a_(n-1) + 4 with a_1 = 7. Always check the difference across at least three pairs. Two pairs can agree by coincidence.

Now take 4, 12, 36, 108. Subtracting gives 8, 24, 72, which is not constant, so it is not arithmetic. Dividing gives 12 / 4 = 3, 36 / 12 = 3, and 108 / 36 = 3. The ratio is constant, so this is geometric with r = 3, and the recursive formula is a_n = a_(n-1) * 3 with a_1 = 4.

1. Write the first term as the base case, a_1.
2. Subtract each term from the one after it. If every result is the same, you have a common difference d.
3. If the differences vary, divide each term by the one before it. If every result is the same, you have a common ratio r.
4. If neither is constant, test whether each term is the sum of the two before it (Fibonacci-style) or whether the multiplier itself grows (factorial-style).
5. Write the rule and verify it by regenerating the original terms from the base case.

Recursive vs explicit (closed-form) formulas

A recursive formula contrasts with an explicit formula, also called a closed-form formula, which gives a_n directly from n without needing any earlier term. For the arithmetic example above, the explicit formula is a_n = 3 + 5(n - 1), so you can find a_100 in one calculation instead of computing ninety-nine terms first.

Both describe the same sequence, but they answer different questions. Use recursion when the relationship between neighbouring terms is the natural idea, such as adding interest each month. Use the explicit form when you want one specific far-off term quickly.

How do you convert a recursive formula to an explicit one?

To convert a recursive formula to an explicit one, identify the pattern the rule creates. If each step adds a constant d, the sequence is arithmetic and its explicit form is a_n = a_1 + (n - 1)d. If each step multiplies by a constant r, the sequence is geometric and its explicit form is a_n = a_1 * r^(n-1). Once you have the explicit form you can find any term in one calculation.

Take the recursive rule a_n = a_(n-1) + 5 with a_1 = 3 from earlier. Because it adds a constant, it is arithmetic with d = 5, so the explicit form is a_n = 3 + 5(n - 1). Check it against the worked terms: for n = 5 this gives 3 + 5(4) = 23, exactly the a_5 we built step by step. That check matters, because the most common conversion error is an off-by-one slip on the exponent or the bracket. If your explicit form does not reproduce a_1, you have used (n) where you needed (n - 1).

You can confirm any term this way with the arithmetic sequence formula or the matching geometric sequence formula for the multiplying case.

Fibonacci-style rules that reach back two terms also have a closed form, but it is far from obvious. The ratio between consecutive Fibonacci terms settles toward the golden ratio, roughly 1.618, and the closed form (Binet's formula) is built from that constant out of powers of the golden ratio divided by the square root of 5. See Wolfram MathWorld on Binet's Fibonacci number formula for the exact statement. This is the clearest sign that recursive and explicit forms are not always equally convenient: the recursion here is trivial to state, while the closed form needs an irrational constant.

Where do recursive formulas show up in real life?

Recursive formulas show up anywhere each state grows out of the one before it: compound interest in finance, population growth in nature, self-calling algorithms in computer science, and running totals in everyday math. Recursion is not just a textbook exercise, it is how many real systems actually update.

• Finance: a savings balance that earns interest each period follows a geometric recursion, the foundation of compound interest. A balance of 1,000 at 5 percent per year is just a_n = a_(n-1) * 1.05.
• Biology and nature: population models where next year's size depends on this year's size, and the branching and spiral patterns often linked to the Fibonacci sequence.
• Computer science: algorithms that call themselves, such as sorting and tree traversal, are built on recursion. The base case is what stops the function calling itself forever.
• Everyday math: loan repayments, depreciation schedules, and any running total that updates one step at a time.

Common mistakes to avoid

• Forgetting the base case. A rule like a_n = a_(n-1) + 5 means nothing until you state a_1. Without it the sequence cannot start.
• Mixing up the index. a_(n-1) is the term right before a_n, not the value n minus 1. Read it as a position, not a number.
• Giving too few starting values. Fibonacci-style rules that reach back two terms need two base cases, a_1 and a_2, not just one.
• Confusing recursive and explicit forms. a_n = a_(n-1) + d is recursive; a_n = a_1 + (n - 1)d is explicit. Do not blend the two in one expression.
• Skipping terms. You cannot jump straight to a_n with a recursive rule. You must build every term in order up to it.
• Assuming every sequence is arithmetic or geometric. Check both the differences and the ratios before deciding, and remember that rules like factorial fit neither family.