Explicit Formula Calculator
Find an explicit formula from sequence terms. The calculator identifies arithmetic or geometric patterns, writes the nth-term rule, compares recursive form, and calculates requested terms.
What an explicit formula is used for
An explicit formula gives the value of a term directly from its position number. If you want the 50th term, you do not have to list the first 49 terms. You place n = 50 into the formula and calculate the answer. That is the main reason explicit formulas are so useful in homework, tests, spreadsheets, and pattern problems.
OpenStax separates explicit and recursive descriptions in its sequence notation lesson. This calculator turns a simple arithmetic or geometric pattern into an explicit formula and also shows the matching recursive form.
Explicit formula patterns
The arithmetic formula uses repeated addition. The geometric formula uses repeated multiplication. In both formulas, n is the term number, a₁ is the first term, d is the common difference, and r is the common ratio.
Worked example: explicit formula for a geometric sequence
Notice that the exponent is n - 1, not n. This is because the first term already has zero multiplications of the ratio applied to it.
Explicit formula vs nth term formula
In many algebra classes, “explicit formula” and “nth term formula” mean almost the same thing. Both describe a rule that gives a term directly from n. A page about explicit formulas can therefore answer informational searches like “how to write an explicit formula,” “how to find the nth term,” and “recursive vs explicit formula.”
Common mistakes to avoid
For arithmetic sequences, students often write aₙ = a₁ + nd instead of aₙ = a₁ + (n - 1)d. For geometric sequences, students often use a₁rⁿ instead of a₁rⁿ⁻¹. Both mistakes shift the sequence by one term. Always test your formula by plugging in n = 1. If it does not return the first term, the formula is wrong.
Common questions
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An explicit formula is a rule that finds a term directly from its term number n. For example, aₙ = 4 + 3(n - 1) lets you find the 20th term by putting n = 20 into the formula. You do not need to calculate every earlier term first.
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First decide whether the sequence is arithmetic or geometric. If it has a constant difference, use aₙ = a₁ + (n - 1)d. If it has a constant ratio, use aₙ = a₁rⁿ⁻¹. Then substitute the first term and the common difference or ratio. Finally, test the formula with the first few terms to make sure it matches the original sequence.
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In many classroom problems, yes. An nth term formula is usually an explicit formula because it gives the value of the nth term directly. Some teachers use the phrase “nth term formula” for arithmetic patterns and “explicit formula” for a broader group of sequences, but the practical idea is the same: use n to find a term.
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The first term is already known as a₁, so the common difference or ratio has been applied zero times at n = 1. That is why arithmetic formulas use (n - 1)d and geometric formulas use rⁿ⁻¹. If you use n instead of n - 1, every term will be shifted because the first term will already be changed once.
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Yes. The first term, common difference, and common ratio can be fractions, decimals, negative numbers, or whole numbers. A sequence such as 1.5, 2.0, 2.5, 3.0 has an explicit arithmetic formula even though the terms are decimals. A sequence such as 1, 1/2, 1/4, 1/8 has an explicit geometric formula with ratio 1/2.
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Then the sequence is not a basic arithmetic or geometric sequence. It may still have an explicit formula, but the formula may involve powers, quadratics, factorials, alternating signs, or another rule. This calculator intentionally focuses on the sequence types most commonly taught first in algebra courses.
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Neither is always better. Recursive formulas are natural when each term depends on the previous term. Explicit formulas are better when you need a far-away term quickly. In homework, the wording matters: if the question asks for a recursive formula, include the starting value and previous-term rule; if it asks for an explicit formula, write a rule in terms of n.
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Plug in n = 1, n = 2, and n = 3. The formula should reproduce the first three terms of the sequence. This quick check catches most off-by-one errors, especially the common mistake of using n instead of n - 1 in the formula.