How do you find a recursive formula from a sequence?

Write down the first term. Then subtract consecutive terms: if the differences are all the same number d, the rule is aₙ = aₙ₋₁ + d. If not, divide consecutive terms: if the ratios are all the same number r, the rule is aₙ = r·aₙ₋₁. If neither works, check the second differences (constant means a quadratic, so add a linear expression in n each step), whether each difference is a fixed multiple of the one before (aₙ = p·aₙ₋₁ + q), or whether each term comes from the two before it (Fibonacci-type). Always state the first term and the values of n the rule applies to.

What is the recursive rule for the sequence 12, 17, 22, 27, …?

Each term is 5 more than the one before, so the sequence is arithmetic with d = 5. The recursive rule is f(1) = 12, f(n) = f(n − 1) + 5 for n ≥ 2 (or a₁ = 12, aₙ = aₙ₋₁ + 5). The explicit formula is aₙ = 5n + 7, so the 10th term is 57 and the next term after 27 is 32.

Which recursive formula generates 6, 1, −4, −9, −14, … where f(1) = 6?

The terms go down by 5 each time, so the answer is f(n + 1) = f(n) − 5 for n ≥ 1. Equivalent forms are f(n) = f(n − 1) − 5 for n ≥ 2 and a₁ = 6, aₙ = aₙ₋₁ − 5. The other choices are wrong: f(n + 1) = f(n) + 5 increases, f(n) = f(n + 1) − 5 also describes an increasing sequence, and f(n + 1) = −5f(n) would give 6, −30, 150, …. The explicit formula is aₙ = −5n + 11.

How do you complete the recursive formula of an arithmetic sequence like −3, −1, 1, 3, …?

Fill in the first term and the common difference. For −3, −1, 1, 3, … : b(1) = −3 and b(n) = b(n − 1) + 2. For −1, −13, −25, −37, … : a(1) = −1 and a(n) = a(n − 1) + (−12), i.e. a(n) = a(n − 1) − 12. For 6, 8, 10, 12, … : a(1) = 6, a(n) = a(n − 1) + 2. For 9, 15, 21, 27, … : a(1) = 9, a(n) = a(n − 1) + 6.

What is the recursive formula for an arithmetic sequence?

a₁ = (first term), aₙ = aₙ₋₁ + d for n ≥ 2, where d is the common difference (any term minus the term before it). Example: with d = 7 and b₁ = 2, the formula is b₁ = 2, bₙ = bₙ₋₁ + 7, giving 2, 9, 16, 23, …. The matching explicit formula is aₙ = a₁ + (n − 1)d — here bₙ = 7n − 5.

What is the recursive formula for a geometric sequence?

a₁ = (first term), aₙ = r·aₙ₋₁ for n ≥ 2, where r is the common ratio (any term divided by the one before). For 2, 6, 18, 54, …: a₁ = 2; aₙ = 3aₙ₋₁, explicit aₙ = 2 · 3ⁿ⁻¹. For 3, −6, 12, −24, 48, …: r = −2, so f(n + 1) = −2f(n) with f(1) = 3, and aₙ = 3 · (−2)ⁿ⁻¹. The matching explicit formula is aₙ = a₁ · rⁿ⁻¹.

Which recursive formula is correct for 80, 40, 20, 10, 5, …?

Each term is half the previous one, so r = 1/2: a₁ = 80, aₙ = (1/2)aₙ₋₁ (equivalently aₙ = aₙ₋₁ ÷ 2 or 0.5aₙ₋₁) for n ≥ 2. It is not arithmetic, because the differences (−40, −20, −10, −5) change. Explicit formula: aₙ = 80 · (1/2)ⁿ⁻¹, so a₈ = 0.625.

How do you write a recursive formula for 16, −8, 4, … or 1/3, 2/9, 4/27, …?

Divide a term by the one before it. 16, −8, 4: r = −8 ÷ 16 = −1/2, so a₁ = 16, aₙ = −(1/2)aₙ₋₁. 1/3, 2/9, 4/27: r = (2/9) ÷ (1/3) = 2/3, so a₁ = 1/3; aₙ = (2/3)aₙ₋₁, explicit aₙ = (1/3) · (2/3)ⁿ⁻¹. 4, −1, 0.25, −0.0625: r = −0.25, so a₁ = 4, aₙ = −0.25aₙ₋₁. A negative ratio makes the signs alternate.

What is the difference between a recursive and an explicit formula?

A recursive formula tells you how to get the next term from previous terms, so you must know earlier terms (a₁ = 12, aₙ = aₙ₋₁ + 5). An explicit formula gives any term from its position directly (aₙ = 5n + 7), so the 100th term is just 5(100) + 7 = 507. Both describe the same sequence. Recursive formulas show the pattern and are natural for spreadsheets and programs; explicit formulas are faster for far-away terms.

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

Identify the type. If the rule adds d, use aₙ = a₁ + (n − 1)d: a₁ = 15, aₙ = aₙ₋₁ + 16 becomes aₙ = 15 + 16(n − 1) = 16n − 1. If the rule multiplies by r, use aₙ = a₁ · rⁿ⁻¹: a₁ = −5, aₙ = −3aₙ₋₁ becomes aₙ = −5 · (−3)ⁿ⁻¹. For aₙ = p·aₙ₋₁ + q, subtract the fixed point L = q/(1 − p) to get a geometric sequence. Write the answer in simplest form by distributing and combining constants.

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

Find the first term by substituting n = 1, then find what changes from one term to the next. aₙ = −9n + 38 gives a₁ = 29 and each term is 9 less: a₁ = 29; aₙ = aₙ₋₁ − 9 (n ≥ 2). aₙ = 4(2)ⁿ⁻¹ gives a₁ = 4 and each term doubles: a₁ = 4; aₙ = 2aₙ₋₁. In a linear explicit formula the coefficient of n is the common difference; in an exponential one the base is the common ratio.

If a₁ = 25 and aₙ = aₙ₋₁ − 2.5, what is a₁₀?

a₁₀ = 25 + 9(−2.5) = 25 − 22.5 = 2.5. The first terms are 25, 22.5, 20, 17.5, 15, … and the explicit formula is aₙ = −2.5n + 27.5.

A sequence has u₁ = 3 and uₙ = 2uₙ₋₁ − 1. What is u₄?

u₂ = 2(3) − 1 = 5, u₃ = 2(5) − 1 = 9, u₄ = 2(9) − 1 = 17, so the answer is 17. The explicit formula is uₙ = 2ⁿ + 1. This rule multiplies and then adds, so the sequence is neither arithmetic nor geometric.

If t₁ = −5 and tₙ₊₁ = tₙ + 3, what are the first five terms, and is 52 a term?

The first five terms are −5, −2, 1, 4, 7. The explicit formula is tₙ = 3n − 8. Setting 3n − 8 = 52 gives n = 20, a whole number, so 52 is the 20th term. A number like 50 would give n = 58/3, so it is not a term.

How do you find a recursive formula from two terms that are far apart?

For an arithmetic sequence, divide the change in value by the change in position. If a₁₈ = 26 and a₃₅ = 60, then d = (60 − 26) ÷ (35 − 18) = 34 ÷ 17 = 2, and a₁ = 26 − 17·2 = −8. So a₁ = −8; aₙ = aₙ₋₁ + 2, and the explicit formula is aₙ = 2n − 10.

If a₃ = 108 and a₅ = 3888 in a geometric sequence, what are the recursive formula and a₈?

a₅ ÷ a₃ = r² = 3888 ÷ 108 = 36, so r = 6 or r = −6, and a₁ = 108 ÷ r² = 3. With r = 6: a₁ = 3; aₙ = 6aₙ₋₁ and a₈ = 839808. With r = −6 the terms alternate in sign and a₈ = −839808. Similarly a₃ = 27, a₄ = 81 gives r = 3, a₁ = 3 and a₈ = 6561.

What is the recursive formula for the Fibonacci sequence?

F₁ = 1, F₂ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 3, giving 1, 1, 2, 3, 5, 8, 13, 21, …; F₁₂ = 144. Common Core writes it as f(0) = f(1) = 1, f(n + 1) = f(n) + f(n − 1) for n ≥ 1. Its explicit form is Binet’s formula, Fₙ = (φⁿ − ψⁿ) ÷ √5, with φ = (1 + √5)/2 ≈ 1.618 (the golden ratio) and ψ = (1 − √5)/2.

What is the recursive formula for 3, 5, 11, 21, 43, 85, …?

Neither differences nor ratios are constant, but each term equals the previous term plus twice the one before it: 11 = 5 + 2·3, 21 = 11 + 2·5, 43 = 21 + 2·11. So a₁ = 3, a₂ = 5; aₙ = aₙ₋₁ + 2aₙ₋₂ for n ≥ 3. The next term is 171, and the explicit formula is aₙ = (8/3) · 2ⁿ⁻¹ + (1/3) · (−1)ⁿ⁻¹.

What is the recursive formula for 1, 3, 6, 10, 15, 21, … (triangular numbers)?

The differences 2, 3, 4, 5, 6 grow by 1, so each step adds the position number: a₁ = 1; aₙ = aₙ₋₁ + n for n ≥ 2. The explicit formula is aₙ = n(n + 1)/2, so a₁₀ = 55. (A list like 1, 3, 6, 10, 15, 21, 28, 36, 47, 57 breaks the pattern at 47 — the triangular numbers continue 45, 55.)

What is the formula for a sequence where you multiply by 2 and add 3?

That rule is aₙ = 2aₙ₋₁ + 3. It is not arithmetic or geometric. Starting at a₁ = 1 it gives 1, 5, 13, 29, 61, …, and the explicit formula is aₙ = 4 · 2ⁿ⁻¹ − 3, found by adding 3 to every term so the sequence becomes geometric.

What is the explicit formula for −2, 2, 6, 10, …?

The common difference is 4 and a₁ = −2, so aₙ = −2 + 4(n − 1) = 4n − 6, which is the choice written aₙ = −6 + 4n. The recursive formula is a₁ = −2, aₙ = aₙ₋₁ + 4.

A sequence is defined by a₁ = 32 and aₙ = (1/4)aₙ₋₁. What is the explicit definition?

This is geometric with a₁ = 32 and r = 1/4, so aₙ = 32(1/4)ⁿ⁻¹, giving 32, 8, 2, 1/2, 1/8, …. Likewise 8, 20, 50, 125, 312.5, … has r = 20 ÷ 8 = 2.5, so aₙ = 8(2.5)ⁿ⁻¹ and recursively a₁ = 8, aₙ = 2.5aₙ₋₁.

What do f(n), aₙ, f(n + 1) and “for n ≥ 2” mean in a recursive formula?

aₙ and f(n) both mean “the nth term”; aₙ₋₁ or f(n − 1) is the term before it. f(n + 1) = f(n) + 5 says the same thing as f(n) = f(n − 1) + 5, just shifted by one. “For n ≥ 2” tells you where the rule starts: the first term is given, and the rule produces terms 2, 3, 4, …. Some books start at a₀ (n = 0); the calculator handles both.

Why does a recursive formula need a first term?

The rule alone only says how terms relate. aₙ = aₙ₋₁ + 5 fits 12, 17, 22, … and also 1, 6, 11, …. The initial condition (a₁ = 12) picks out one sequence. Rules that use two previous terms, like Fibonacci, need two starting values.

How does this recursive formula calculator work?

It reads your terms (integers, decimals, fractions or LaTeX like \frac{1}{3}) and tests them in order: constant, arithmetic (equal differences), geometric (equal ratios), polynomial up to degree 4 (finite differences), aₙ = p·aₙ₋₁ + q, and aₙ = p·aₙ₋₁ + q·aₙ₋₂. The first pattern that reproduces every term you entered gives the recursive rule in subscript and function notation, the explicit formula (solved exactly, or via the characteristic equation for two-term rules), any term, the sum, a table and steps. The other tabs start from a rule, an explicit formula or two known terms.

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Algebra 1 · Algebra 2 · Precalculus · updated October 2, 2026

Recursive Formula Calculator: Find the Recursive Rule of Any Sequence

Quick answer

A recursive formula gives the first term and a rule that builds each term from the one before. Arithmetic sequence: a₁ = first term, aₙ = aₙ₋₁ + d. Geometric sequence: aₙ = r·aₙ₋₁. Example: 12, 17, 22, 27, … adds 5 each time, so the recursive formula is a₁ = 12, aₙ = aₙ₋₁ + 5 for n ≥ 2 (in function notation f(1) = 12, f(n) = f(n − 1) + 5). The explicit formula is aₙ = 5n + 7, so the 10th term is 57.

Paste any sequence — whole numbers, decimals, fractions or negatives — and get the recursive formula in aₙ and f(n) notation, the explicit formula, any term, the sum and a step-by-step explanation. It recognizes arithmetic, geometric, quadratic, multiply-then-add and Fibonacci-type patterns, and it also works backwards from a recursive rule, an explicit formula or two known terms.

Recursive formula from any list of terms
aₙ, f(n) and f(n + 1) notation, from a₁ or a₀
Explicit ↔ recursive conversion and nth term
Two-term problems and “is this a term?” checks

Last updated October 2, 2026. Free, no sign-up. Guides: recursive formula for an arithmetic sequence · recursive vs explicit formula

Recursive Formula Calculator

Paste a sequence to get its recursive rule, explicit formula and any term — or start from a recursive rule, an explicit formula or two known terms.

Sequence terms

Separate terms with commas or spaces. Fractions (1/3), decimals and negatives are fine; “…” is ignored. Enter 4 or more terms for the most reliable answer.

Find term number n

First term is

Try an example

Arithmetic sequence

Recursive formula

a₁ = 12

aₙ = aₙ₋₁ + 5 for n ≥ 2

Explicit formula (nth term)

aₙ = 5n + 7

Also: aₙ = 12 + 5(n − 1)

Term 10

a₁₀ = 57

Sum of terms a₁ to a₁₀: S₁₀ = 345

Next 5 terms

32, 37, 42, 47, 52

Parameters

a₁ = 12, d = 5

a₁: first term · d: common difference

Same formula in other notations

f(1) = 12; f(n) = f(n − 1) + 5 for n ≥ 2
f(1) = 12; f(n + 1) = f(n) + 5 for n ≥ 1

n	aₙ	aₙ − aₙ₋₁	aₙ ÷ aₙ₋₁
1	12	—	—
2	17	5	17/12
3	22	5	22/17
4	27	5	27/22
5	32	5	32/27
6	37	5	1.15625
7	42	5	42/37
8	47	5	47/42

Shaded rows are the terms you entered; the rest are generated by the formula.

Step by step

Subtract each term from the next: 17 − 12 = 5, 22 − 17 = 5, 27 − 22 = 5. The differences are all 5, so the sequence is arithmetic with common difference d = 5.
Recursive formula = starting value(s) + rule: a₁ = 12; aₙ = aₙ₋₁ + 5, for n ≥ 2. In function notation: f(1) = 12; f(n) = f(n − 1) + 5.
Explicit formula: aₙ = a₁ + (n − 1)d = 12 + 5(n − 1) = 5n + 7.
Term 10: a₁₀ = 12 + (10 − 1)(5) = 57.

How to find a recursive formula

A recursive formula defines a sequence by giving its first term (or first few terms) and a rule that produces each later term from the term or terms before it. For an arithmetic sequence the rule adds a common difference d (aₙ = aₙ₋₁ + d); for a geometric sequence it multiplies by a common ratio r (aₙ = r·aₙ₋₁). An explicit formula, by contrast, computes any term directly from its position n.

Arithmetic: a₁ given, aₙ = aₙ₋₁ + d → aₙ = a₁ + (n − 1)d
Geometric: a₁ given, aₙ = r·aₙ₋₁ → aₙ = a₁ · rⁿ⁻¹
d = aₙ − aₙ₋₁ · r = aₙ ÷ aₙ₋₁

First term. For 12, 17, 22, 27 it is a₁ = 12.
Differences. 17 − 12 = 5, 22 − 17 = 5, 27 − 22 = 5. All equal, so the sequence is arithmetic with d = 5.
Write the rule with its starting value: a₁ = 12, aₙ = aₙ₋₁ + 5 for n ≥ 2.
Ratios, if the differences vary. For 2, 6, 18, 54 each ratio is 3, giving a₁ = 2; aₙ = 3aₙ₋₁.
Other patterns. Constant second differences (1, 3, 6, 10, …) give aₙ = aₙ₋₁ + n; growing differences that double (3, 5, 9, 17, …) give aₙ = 2aₙ₋₁ − 1; terms built from the two before (1, 1, 2, 3, 5, …) give aₙ = aₙ₋₁ + aₙ₋₂.
Check and convert. Regenerate the terms from your rule. For far-away terms use the explicit formula: aₙ = 5n + 7, so a₁₀₀ = 507.

Why the starting value matters: the rule aₙ = aₙ₋₁ + 5 fits 12, 17, 22, … and also 1, 6, 11, …. Only the rule plus a₁ pins down one sequence. A rule that uses two earlier terms needs two starting values.

Recursive formulas for common sequences

Answers for the sequences students search for most, computed by the calculator above. “Next” is the term after the last one shown.

Sequence	Type	Recursive formula	Explicit formula	Next
12, 17, 22, 27, 32, 37, …	Arithmetic	a₁ = 12; aₙ = aₙ₋₁ + 5	aₙ = 5n + 7	32
6, 1, −4, −9, −14, −19, …	Arithmetic	a₁ = 6; aₙ = aₙ₋₁ − 5	aₙ = −5n + 11	−19
−3, −1, 1, 3, 5, 7, …	Arithmetic	a₁ = −3; aₙ = aₙ₋₁ + 2	aₙ = 2n − 5	5
−1, −13, −25, −37, −49, −61, …	Arithmetic	a₁ = −1; aₙ = aₙ₋₁ − 12	aₙ = −12n + 11	−49
1, 6, 11, 16, 21, 26, …	Arithmetic	a₁ = 1; aₙ = aₙ₋₁ + 5	aₙ = 5n − 4	21
6, 8, 10, 12, 14, 16, …	Arithmetic	a₁ = 6; aₙ = aₙ₋₁ + 2	aₙ = 2n + 4	14
9, 15, 21, 27, 33, 39, …	Arithmetic	a₁ = 9; aₙ = aₙ₋₁ + 6	aₙ = 6n + 3	33
19, 15, 11, 7, 3, −1, …	Arithmetic	a₁ = 19; aₙ = aₙ₋₁ − 4	aₙ = −4n + 23	3
−42, −33, −24, −15, −6, 3, …	Arithmetic	a₁ = −42; aₙ = aₙ₋₁ + 9	aₙ = 9n − 51	−6
5, 3, 1, −1, −3, −5, …	Arithmetic	a₁ = 5; aₙ = aₙ₋₁ − 2	aₙ = −2n + 7	−3
2, 6, 18, 54, 162, 486, …	Geometric	a₁ = 2; aₙ = 3aₙ₋₁	aₙ = 2 · 3ⁿ⁻¹	162
80, 40, 20, 10, 5, 2.5, …	Geometric	a₁ = 80; aₙ = 0.5aₙ₋₁	aₙ = 80 · (0.5)ⁿ⁻¹	2.5
16, −8, 4, −2, 1, −0.5, …	Geometric	a₁ = 16; aₙ = −0.5aₙ₋₁	aₙ = 16 · (−0.5)ⁿ⁻¹	−2
3, −6, 12, −24, 48, −96, …	Geometric	a₁ = 3; aₙ = −2aₙ₋₁	aₙ = 3 · (−2)ⁿ⁻¹	−96
5, 15, 45, 135, 405, 1215, …	Geometric	a₁ = 5; aₙ = 3aₙ₋₁	aₙ = 5 · 3ⁿ⁻¹	1215
3, 27, 243, 2187, 19683, 177147, …	Geometric	a₁ = 3; aₙ = 9aₙ₋₁	aₙ = 3 · 9ⁿ⁻¹	19683
8, 40, 200, 1000, 5000, 25000, …	Geometric	a₁ = 8; aₙ = 5aₙ₋₁	aₙ = 8 · 5ⁿ⁻¹	5000
150, 30, 6, 1.2, 0.24, 0.048, …	Geometric	a₁ = 150; aₙ = 0.2aₙ₋₁	aₙ = 150 · (0.2)ⁿ⁻¹	1.2
1/3, 2/9, 4/27, 8/81, 16/243, 32/729, …	Geometric	a₁ = 1/3; aₙ = (2/3)aₙ₋₁	aₙ = (1/3) · (2/3)ⁿ⁻¹	8/81
4, −1, 0.25, −0.0625, 0.015625, −1/256, …	Geometric	a₁ = 4; aₙ = −0.25aₙ₋₁	aₙ = 4 · (−0.25)ⁿ⁻¹	0.015625
−0.25, −2, −16, −128, −1024, −8192, …	Geometric	a₁ = −0.25; aₙ = 8aₙ₋₁	aₙ = −0.25 · 8ⁿ⁻¹	−1024
1, 3, 6, 10, 15, 21, …	Quadratic+	a₁ = 1; aₙ = aₙ₋₁ + n	aₙ = (1/2)n² + (1/2)n	28
3, 5, 9, 17, 33, 65, …	aₙ = p·aₙ₋₁ + q	a₁ = 3; aₙ = 2aₙ₋₁ − 1	aₙ = 2 · 2ⁿ⁻¹ + 1	65
1, 1, 2, 3, 5, 8, …	Two-term	a₁ = 1, a₂ = 1; aₙ = aₙ₋₁ + aₙ₋₂	aₙ ≈ 0.7236 · (1.618)ⁿ⁻¹ + 0.2764 · (−0.618)ⁿ⁻¹	13
3, 5, 11, 21, 43, 85, …	Two-term	a₁ = 3, a₂ = 5; aₙ = aₙ₋₁ + 2aₙ₋₂	aₙ = (8/3) · 2ⁿ⁻¹ + (1/3) · (−1)ⁿ⁻¹	171

Recursive formula cheat sheet

Pattern	How to spot it	Recursive formula	Explicit formula	Example
Arithmetic	Same difference d	a₁ given; aₙ = aₙ₋₁ + d	aₙ = a₁ + (n − 1)d	12, 17, 22, 27 (d = 5)
Geometric	Same ratio r	a₁ given; aₙ = r·aₙ₋₁	aₙ = a₁ · rⁿ⁻¹	2, 6, 18, 54 (r = 3)
Quadratic	Same second difference	aₙ = aₙ₋₁ + (linear in n)	aₙ = An² + Bn + C	1, 3, 6, 10, 15 (aₙ = aₙ₋₁ + n)
Multiply-then-add	Differences form a geometric sequence	aₙ = p·aₙ₋₁ + q	aₙ = (a₁ − L)pⁿ⁻¹ + L, L = q/(1 − p)	3, 5, 9, 17 (aₙ = 2aₙ₋₁ − 1)
Fibonacci-type	Each term from the two before	a₁, a₂ given; aₙ = p·aₙ₋₁ + q·aₙ₋₂	Roots of x² = px + q (Binet for Fibonacci)	1, 1, 2, 3, 5, 8

Recursive to explicit: worked conversions

“Use the initial term and the recursive formula to find an explicit formula” — answers in simplest form, with the 10th term.

Recursive	Explicit	First terms	10th term
a₁ = 15; aₙ = aₙ₋₁ + 16	aₙ = 16n − 1 (= 15 + 16(n − 1))	15, 31, 47, 63, 79	a₁₀ = 159
a₁ = −37; aₙ = aₙ₋₁ − 16	aₙ = −16n − 21 (= −37 − 16(n − 1))	−37, −53, −69, −85, −101	a₁₀ = −181
a₁ = −23; aₙ = aₙ₋₁ + 17	aₙ = 17n − 40 (= −23 + 17(n − 1))	−23, −6, 11, 28, 45	a₁₀ = 130
a₁ = 28; aₙ = aₙ₋₁ + 10	aₙ = 10n + 18 (= 28 + 10(n − 1))	28, 38, 48, 58, 68	a₁₀ = 118
a₁ = 2; aₙ = aₙ₋₁ + 7	aₙ = 7n − 5 (= 2 + 7(n − 1))	2, 9, 16, 23, 30	a₁₀ = 65
a₁ = 25; aₙ = aₙ₋₁ − 2.5	aₙ = −2.5n + 27.5 (= 25 − 2.5(n − 1))	25, 22.5, 20, 17.5, 15	a₁₀ = 2.5
a₁ = −14; aₙ = aₙ₋₁ + 6	aₙ = 6n − 20 (= −14 + 6(n − 1))	−14, −8, −2, 4, 10	a₁₀ = 40
a₁ = −5; aₙ = −3aₙ₋₁	aₙ = −5 · (−3)ⁿ⁻¹	−5, 15, −45, 135, −405	a₁₀ = 98415
a₁ = 185; aₙ = (2/3)aₙ₋₁	aₙ = 185 · (2/3)ⁿ⁻¹	185, 370/3, 740/9, 1480/27, 2960/81	a₁₀ = 4.8122746
a₁ = 32; aₙ = (1/4)aₙ₋₁	aₙ = 32 · (1/4)ⁿ⁻¹	32, 8, 2, 1/2, 1/8	a₁₀ = 1/8192
a₁ = 3; aₙ = 2aₙ₋₁ − 1	aₙ = 2 · 2ⁿ⁻¹ + 1 (= 2ⁿ + 1)	3, 5, 9, 17, 33	a₁₀ = 1025
a₁ = 1; aₙ = 2aₙ₋₁ + 3	aₙ = 4 · 2ⁿ⁻¹ − 3	1, 5, 13, 29, 61	a₁₀ = 2045

Explicit to recursive: worked conversions

Substitute n = 1 for the first term, then find the common difference or ratio.

Explicit	First terms	Recursive
aₙ = −9n + 38	29, 20, 11, 2	a₁ = 29; aₙ = aₙ₋₁ − 9
aₙ = 5n + 7	12, 17, 22, 27	a₁ = 12; aₙ = aₙ₋₁ + 5
aₙ = −2 + 4n	2, 6, 10, 14	a₁ = 2; aₙ = aₙ₋₁ + 4
aₙ = 9 + (n−1)(−3)	9, 6, 3, 0	a₁ = 9; aₙ = aₙ₋₁ − 3
aₙ = 4(2)ⁿ⁻¹	4, 8, 16, 32	a₁ = 4; aₙ = 2aₙ₋₁
aₙ = 15(1/3)ⁿ⁻¹	15, 5, 5/3, 5/9	a₁ = 15; aₙ = (1/3)aₙ₋₁
aₙ = 32(1/4)ⁿ⁻¹	32, 8, 2, 1/2	a₁ = 32; aₙ = (1/4)aₙ₋₁
aₙ = 8(2.5)ⁿ⁻¹	8, 20, 50, 125	a₁ = 8; aₙ = 2.5aₙ₋₁
aₙ = −5(−3)ⁿ⁻¹	−5, 15, −45, 135	a₁ = −5; aₙ = −3aₙ₋₁
aₙ = n² + 1	2, 5, 10, 17	a₁ = 2; aₙ = aₙ₋₁ + (2n − 1)
aₙ = 2ⁿ + 1	3, 5, 9, 17	a₁ = 3; aₙ = 2aₙ₋₁ − 1

Finding the formula from two terms

Arithmetic: d = (aⱼ − aᵢ) ÷ (j − i). Geometric: rʲ⁻ⁱ = aⱼ ÷ aᵢ — when the gap is even there are two real answers, r and −r. Then work back to a₁. The last column is a₈.

Given	d or r	Recursive formula	Explicit formula	a₈
a₁₈ = 26, a₃₅ = 60	d = 2	a₁ = −8; aₙ = aₙ₋₁ + 2	aₙ = 2n − 10	a₈ = 6
a₄ = 25, a₁₀ = 43	d = 3	a₁ = 16; aₙ = aₙ₋₁ + 3	aₙ = 3n + 13	a₈ = 37
a₃ = 108, a₅ = 3888	r = 6 or −6	a₁ = 3; aₙ = 6aₙ₋₁	aₙ = 3 · 6ⁿ⁻¹	a₈ = 839808 or −839808
a₅ = −324, a₆ = 972	r = −3	a₁ = −4; aₙ = −3aₙ₋₁	aₙ = −4 · (−3)ⁿ⁻¹	a₈ = 8748
a₃ = 27, a₄ = 81	r = 3	a₁ = 3; aₙ = 3aₙ₋₁	aₙ = 3 · 3ⁿ⁻¹	a₈ = 6561
a₅ = 512, a₆ = 2048	r = 4	a₁ = 2; aₙ = 4aₙ₋₁	aₙ = 2 · 4ⁿ⁻¹	a₈ = 32768

Recursive formula notation: aₙ, f(n) and f(n + 1)

Textbooks and tests write the same rule several ways. All of these describe 12, 17, 22, 27, …. Multiple-choice questions often switch between them, so match the starting index and the direction of the shift.

Style	Starting value	Rule	Applies
Subscript	a₁ = 12	aₙ = aₙ₋₁ + 5	for n ≥ 2
Function	f(1) = 12	f(n) = f(n − 1) + 5	for n ≥ 2
Function, next term	f(1) = 12	f(n + 1) = f(n) + 5	for n ≥ 1
Subscript, next term	a₁ = 12	aₙ₊₁ = aₙ + 5	for n ≥ 1
Starting at n = 0	a₀ = 12	aₙ = aₙ₋₁ + 5	for n ≥ 1 (explicit: aₙ = 5n + 12)

Recursive = step by step

You need the previous term to get the next one. Great for spreadsheets, code and seeing how a pattern grows, slow for the 1,000th term.

Explicit = jump anywhere

aₙ = 5n + 7 gives a₁₀₀₀ = 5007 instantly. Linear explicit formulas are arithmetic; exponential ones are geometric.

Same sequence, two views

Arithmetic sequences are linear functions of n with slope d; geometric sequences are exponential functions with base r.

Common mistakes

Leaving out the first term. aₙ = aₙ₋₁ + 5 alone is not a complete recursive formula.
Subtracting in the wrong order: d = later term − earlier term. For 6, 1, −4 it is 1 − 6 = −5, not +5.
Calling 80, 40, 20, 10 arithmetic. The differences change; the ratio (1/2) is constant, so it is geometric.
Mixing up f(n + 1) = f(n) − 5 and f(n) = f(n + 1) − 5. The second one describes an increasing sequence.
Typing 4(2)^n-1 when you mean 4(2)^(n−1). Without parentheses the −1 is subtracted after the power.
Taking only the positive root when two geometric terms are an even number of steps apart: r² = 36 means r = 6 or r = −6.

Sources and how this calculator was checked

Definitions and formulas follow the OpenStax College Algebra and Precalculus textbooks and Paul’s Online Math Notes; the two-term and characteristic-equation methods follow MathWorld and standard references on linear recurrences, and the topic matches Common Core standards HSF-BF.A.2 and HSF-IF.A.3. Each detected pattern must reproduce every term you enter exactly (to floating-point tolerance), fractions are recovered exactly, and the Fibonacci and triangular-number cases were checked against OEIS. Every table on this page is generated by the same code as the calculator.

OpenStax College Algebra 2e — 9.1 Sequences and Their NotationsRecursive formulas, initial terms, explicit formulas and factorial notation.
OpenStax College Algebra 2e — 9.2 Arithmetic SequencesRecursive formula aₙ = aₙ₋₁ + d and explicit formula aₙ = a₁ + (n − 1)d.
OpenStax College Algebra 2e — 9.3 Geometric SequencesRecursive formula aₙ = r·aₙ₋₁ and explicit formula aₙ = a₁rⁿ⁻¹.
OpenStax Precalculus 2e — 11.1 Sequences and Their NotationsWriting terms from recursive and explicit formulas, including the Fibonacci sequence.
OpenStax Precalculus 2e — 11.2 Arithmetic SequencesFinding d and a₁ from two terms and converting between forms.
OpenStax Precalculus 2e — 11.3 Geometric SequencesFinding r from two terms, including the ± case for even gaps.
Paul’s Online Math Notes — SequencesSequence notation, recursive definitions and starting indices.
Wolfram MathWorld — Recurrence EquationDefinition of recurrence relations and linear recurrences with constant coefficients.
Wolfram MathWorld — Binet’s Fibonacci Number FormulaClosed form Fₙ = (φⁿ − ψⁿ)/√5.
Wolfram MathWorld — Finite DifferenceWhy constant kth differences mean a degree-k polynomial.
Wikipedia — Linear recurrence with constant coefficientsCharacteristic equation method for aₙ = p·aₙ₋₁ + q·aₙ₋₂, including repeated and complex roots.
OEIS A000045 — Fibonacci numbersReference values used to test the Fibonacci-type detection.
OEIS A000217 — Triangular numbersReference values for the quadratic example 1, 3, 6, 10, 15, ….
Khan Academy — SequencesLessons on recursive and explicit formulas for arithmetic and geometric sequences.
Common Core State Standards — HSF-BF.A.2Write arithmetic and geometric sequences both recursively and with an explicit formula, and translate between the two forms.
Common Core State Standards — HSF-IF.A.3Sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

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Explicit to recursive: worked conversions

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Recursive formula notation: aₙ, f(n) and f(n + 1)

Common mistakes

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Frequently Asked Questions

What is a recursive formula?

How do you find a recursive formula from a sequence?

What is the recursive rule for the sequence 12, 17, 22, 27, …?

Which recursive formula generates 6, 1, −4, −9, −14, … where f(1) = 6?

How do you complete the recursive formula of an arithmetic sequence like −3, −1, 1, 3, …?

What is the recursive formula for an arithmetic sequence?

What is the recursive formula for a geometric sequence?

Which recursive formula is correct for 80, 40, 20, 10, 5, …?

How do you write a recursive formula for 16, −8, 4, … or 1/3, 2/9, 4/27, …?

What is the difference between a recursive and an explicit formula?

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

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

If a₁ = 25 and aₙ = aₙ₋₁ − 2.5, what is a₁₀?

A sequence has u₁ = 3 and uₙ = 2uₙ₋₁ − 1. What is u₄?

If t₁ = −5 and tₙ₊₁ = tₙ + 3, what are the first five terms, and is 52 a term?

How do you find a recursive formula from two terms that are far apart?

If a₃ = 108 and a₅ = 3888 in a geometric sequence, what are the recursive formula and a₈?

What is the recursive formula for the Fibonacci sequence?

What is the recursive formula for 3, 5, 11, 21, 43, 85, …?

What is the recursive formula for 1, 3, 6, 10, 15, 21, … (triangular numbers)?

What is the formula for a sequence where you multiply by 2 and add 3?

What is the explicit formula for −2, 2, 6, 10, …?

A sequence is defined by a₁ = 32 and aₙ = (1/4)aₙ₋₁. What is the explicit definition?

What do f(n), aₙ, f(n + 1) and “for n ≥ 2” mean in a recursive formula?

Why does a recursive formula need a first term?

How does this recursive formula calculator work?

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Recursive Formula Calculator: Find the Recursive Rule of Any Sequence

How to find a recursive formula

Recursive formulas for common sequences

Recursive formula cheat sheet

Recursive to explicit: worked conversions

Explicit to recursive: worked conversions

Finding the formula from two terms

Recursive formula notation: aₙ, f(n) and f(n + 1)

Common mistakes

Sources and how this calculator was checked

Frequently Asked Questions

What is a recursive formula?

How do you find a recursive formula from a sequence?

What is the recursive rule for the sequence 12, 17, 22, 27, …?

Which recursive formula generates 6, 1, −4, −9, −14, … where f(1) = 6?

How do you complete the recursive formula of an arithmetic sequence like −3, −1, 1, 3, …?

What is the recursive formula for an arithmetic sequence?

What is the recursive formula for a geometric sequence?

Which recursive formula is correct for 80, 40, 20, 10, 5, …?

How do you write a recursive formula for 16, −8, 4, … or 1/3, 2/9, 4/27, …?

What is the difference between a recursive and an explicit formula?

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

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

If a₁ = 25 and aₙ = aₙ₋₁ − 2.5, what is a₁₀?

A sequence has u₁ = 3 and uₙ = 2uₙ₋₁ − 1. What is u₄?

If t₁ = −5 and tₙ₊₁ = tₙ + 3, what are the first five terms, and is 52 a term?

How do you find a recursive formula from two terms that are far apart?

If a₃ = 108 and a₅ = 3888 in a geometric sequence, what are the recursive formula and a₈?

What is the recursive formula for the Fibonacci sequence?

What is the recursive formula for 3, 5, 11, 21, 43, 85, …?

What is the recursive formula for 1, 3, 6, 10, 15, 21, … (triangular numbers)?

What is the formula for a sequence where you multiply by 2 and add 3?

What is the explicit formula for −2, 2, 6, 10, …?

A sequence is defined by a₁ = 32 and aₙ = (1/4)aₙ₋₁. What is the explicit definition?

What do f(n), aₙ, f(n + 1) and “for n ≥ 2” mean in a recursive formula?

Why does a recursive formula need a first term?

How does this recursive formula calculator work?

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