Recursive Formula Calculator
Write a recursive formula from sequence terms. The calculator checks constant differences and constant ratios, then gives the recursive rule, explicit formula, generated terms, and step-by-step reasoning.
What a recursive formula actually tells you
A recursive formula tells you how to get the next term from the term before it. That is why it normally needs two pieces of information: the starting term and the rule for moving forward. For an arithmetic sequence, the rule adds the same number each time. For a geometric sequence, the rule multiplies by the same number each time.
OpenStax introduces this exact idea in its lesson on sequences and their notations, where sequences can be described by explicit rules or recursive rules. This calculator focuses on the common classroom cases: arithmetic and geometric recursive formulas.
Recursive formula patterns
The important difference is the operation. If the sequence changes by adding or subtracting the same amount, use the arithmetic rule. If the sequence changes by multiplying or dividing by the same amount, use the geometric rule.
Worked example: write a recursive formula
This is a stronger answer than just saying “add 4.” A proper recursive formula states the first term and the rule. Without the first term, the same rule could describe many different sequences.
When recursive formulas are useful
Recursive formulas are especially useful when each new value depends naturally on the previous value: savings that grow each month, populations that multiply, patterns that repeat a fixed change, and computer algorithms that build one result from the last result. They are less convenient when you need the 100th term immediately, because you may need to generate many earlier terms first. In that case, an explicit formula is usually faster.
Common mistakes with recursive formulas
The most common mistake is forgetting to include the starting value. Another mistake is writing an explicit formula when the question asks for a recursive one. A third mistake is assuming every list of numbers is arithmetic or geometric. If the differences and ratios are not constant, a simple recursive formula may still exist, but it will not be one of these two basic patterns.
Common questions
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A recursive formula is a rule that defines a term by using one or more previous terms. In a basic sequence problem, it usually tells you the first term and then gives a rule for finding the next term. For example, a₁ = 3 and aₙ = aₙ₋₁ + 5 means the sequence starts at 3 and adds 5 each time. The first few terms are 3, 8, 13, 18, and so on.
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First check how the terms change. If the same number is added each time, the sequence is arithmetic and the recursive rule is aₙ = aₙ₋₁ + d. If the same number is multiplied each time, the sequence is geometric and the recursive rule is aₙ = r × aₙ₋₁. Then include the first term. A recursive rule is incomplete without a starting value.
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A recursive formula uses the previous term to find the next term. An explicit formula uses the term number n to find a term directly. For example, a recursive formula may say aₙ = aₙ₋₁ + 4, while an explicit formula may say aₙ = 2 + 4(n - 1). Recursive formulas are easier to see from repeated patterns, but explicit formulas are faster when you need a far-away term.
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Many sequences can be written recursively, but not every sequence has a simple arithmetic or geometric recursive rule. This calculator checks the two most common school patterns: constant difference and constant ratio. If your sequence does not match either one, you may need a custom recursive rule, a second-order rule, or a formula involving n.
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The initial term anchors the sequence. The rule tells you how to move from one term to the next, but it does not tell you where to start. For example, aₙ = aₙ₋₁ + 3 could describe 1, 4, 7, 10 or 10, 13, 16, 19. The rule is the same, but the sequences are different because the first term is different.
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Look at the operation between consecutive terms. If subtracting consecutive terms gives the same difference every time, it is arithmetic. If dividing consecutive terms gives the same ratio every time, it is geometric. Some sequences have neither pattern, so checking both differences and ratios is safer than guessing from the first two terms only.
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Yes. Arithmetic sequences can have negative common differences, such as 20, 15, 10, 5. Geometric sequences can have negative ratios, such as 2, -6, 18, -54. The formulas still work, but negative ratios cause signs to alternate from term to term.
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It shows the explicit formula because students often need to compare both forms. The recursive formula explains how the sequence grows step by step, while the explicit formula is better for finding a specific term quickly. Seeing both forms helps you understand the same sequence from two different angles.