Identify the pattern in a number sequence, find the next terms and the nth term formula, and calculate partial sums for arithmetic, geometric, Fibonacci, quadratic and other sequences — a full sequence solver and numerical pattern finder in one tool.
Quick Answer: How Do You Find the Next Number in a Sequence?
Look at the differences between consecutive terms. If the difference is constant, it's an arithmetic sequence — add that difference to get the next term. If the ratio between terms is constant, it's geometric — multiply by that ratio. If each term is the sum of the two before it, it's Fibonacci-like. If neither the difference nor the ratio is constant, check the second differences — if those are constant, the sequence is quadratic. The calculator below runs through all of these checks automatically and extends the sequence as far as you need.
Identify & Extend a Sequence
Enter the known terms of your sequence (at least 3 — 4 or more gives more reliable pattern detection), separated by commas.
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terms
Generate a sequence
Type
First term (a₁)
Common diff/ratio
Number of terms
Analysis Result
Sequence Type
Arithmetic
aₙ = 2 + (n−1)×3
Property
Value
Extended Sequence
Term Table
n
Term (aₙ)
Partial Sum (Sₙ)
Types of Number Sequences
Type
Definition
nth Term Formula
Example
Arithmetic
Constant difference (d) between terms
aₙ = a₁ + (n−1)d
2, 5, 8, 11, 14, …
Geometric
Constant ratio (r) between terms
aₙ = a₁ × rⁿ⁻¹
2, 6, 18, 54, …
Fibonacci-like
Each term = sum of the previous two
aₙ = aₙ₋₁ + aₙ₋₂
1, 1, 2, 3, 5, 8, …
Quadratic
Constant second difference
aₙ = an² + bn + c
2, 4, 7, 11, 16, … (2nd diff = 1)
Triangular
Count of dots forming a triangle
aₙ = n(n+1)/2
1, 3, 6, 10, 15, …
Square numbers
Perfect squares
aₙ = n²
1, 4, 9, 16, 25, …
💡 Did You Know? Every quadratic sequence has a constant second difference equal to twice the coefficient of n² in its formula. For 1, 4, 9, 16, 25 (aₙ = n²), the differences are 3, 5, 7, 9 and the second differences are all 2 — exactly 2 × 1, since the coefficient of n² is 1.
Quick Quiz: Is It Arithmetic, Geometric, or Neither?
A common exam-style question asks you to identify which of several sequences is arithmetic, which is geometric, and which is neither. Work through these four before checking the answers.
Which of the following sequences are arithmetic or geometric?
A. 4, 9, 14, 19, 24, …
B. 3, 6, 12, 24, 48, …
C. 1, 4, 9, 16, 25, …
D. 5, −10, 20, −40, …
Show answers
A is arithmetic — common difference d = 5. B is geometric — common ratio r = 2. C is neither — it's a quadratic sequence of perfect squares (constant second difference of 2), not arithmetic or geometric. D is geometric — common ratio r = −2; a geometric sequence can have a negative ratio, which makes the terms alternate in sign.
How to Find the Next Number in a Sequence
There is a reliable order of checks that covers almost every sequence you will meet in a maths class or a puzzle. First, calculate the differences between consecutive terms — if they're all equal, you have an arithmetic sequence and the pattern is simply "add d each time." Second, if the differences aren't constant, calculate the ratios instead — a constant ratio means a geometric sequence, where the pattern is "multiply by r each time." Third, check whether each term is the sum of the two before it, which identifies a Fibonacci-like sequence. Fourth, if none of those match, take the differences of the differences — the "second differences." If those are constant, the sequence is quadratic, and you can keep extending it by continuing the second-difference pattern even without knowing the full formula.
Nth Term Formulas
Arithmetic: aₙ = a₁ + (n − 1)d
Geometric: aₙ = a₁ × rⁿ⁻¹
Fibonacci-like: aₙ = aₙ₋₁ + aₙ₋₂ a₁ = first term | d = common difference | r = common ratio | n = term position
Sequence: 8, 4, 2, 1, 0.5, … with r = 0.5 (since |r| < 1, the sum converges).
S∞ = a₁ / (1 − r) = 8 / (1 − 0.5) = 8 / 0.5 = 16
This is the same idea behind Zeno's paradox and the repeating-decimal identity 0.999… = 1.
Choosing the Right Pattern: Arithmetic vs Geometric vs Quadratic
Signal
Sequence type
How to extend
Constant difference between terms
Arithmetic
Add the common difference d
Constant ratio between terms
Geometric
Multiply by the common ratio r
Each term = sum of previous two
Fibonacci-like
Add the two preceding terms
Constant second difference
Quadratic
Continue the difference pattern one level down
None of the above
Unrecognised / needs more terms
Provide more known terms, or check for a non-numeric rule
Common Mistakes
Mistake
Why it goes wrong
How to avoid it
Assuming a pattern from only 2 terms
Infinitely many sequences pass through any two points
Use at least 3 terms for arithmetic/geometric, and 4+ for quadratic detection
Mixing up arithmetic and geometric
Adding when you should multiply (or vice versa) gives a completely different sequence
Check differences first, then ratios, before assuming a pattern
Forgetting a geometric ratio can be negative or a fraction
Sequences like 4, −2, 1, −0.5 are geometric with r = −0.5, easy to miss
Divide consecutive terms rather than eyeballing the pattern
Applying the infinite sum formula when |r| ≥ 1
The series diverges — there is no finite sum
Only use S∞ = a₁/(1−r) when the ratio's absolute value is less than 1
Treating every non-arithmetic, non-geometric sequence as "random"
Many such sequences are quadratic, cubic, or Fibonacci-like and are still fully predictable
Check second differences before giving up on finding a pattern
⚠️ Common Mistake: Confusing the sequence 1, 1, 2, 3, 5, 8 (Fibonacci — each term is the sum of the previous two) with a quadratic sequence. Fibonacci-like growth is exponential in the long run, not quadratic, even though early terms can look similar.
Practical Applications
Finance
Compound interest is a geometric sequence — each period's balance is the previous balance multiplied by the same factor (1 + rate). Loan amortisation schedules and superannuation balance projections are built on exactly this pattern.
Computer science
Algorithm growth rates, recursive data structures and time-complexity analysis frequently use arithmetic, geometric and Fibonacci-like sequences — for example, the number of operations in certain divide-and-conquer algorithms follows a geometric pattern.
Nature and biology
The Fibonacci sequence appears throughout nature: the spiral patterns of shells and sunflower seed heads, and the branching of trees, all approximate Fibonacci-like growth, connected to the golden ratio φ ≈ 1.618 that consecutive Fibonacci ratios converge towards.
Education and puzzles
Number sequence questions are a staple of numeracy tests, IQ tests and maths curricula precisely because they test pattern recognition — the same skill used here to classify a sequence as arithmetic, geometric, quadratic or otherwise.
Hard Number Sequences to Test Yourself
Not every sequence is arithmetic, geometric, Fibonacci or quadratic — some of the hardest number sequences in IQ tests and maths competitions combine two rules at once, or use a rule that only becomes obvious once you look at ratios and differences together. Try these before checking the answers, then paste them into the calculator above to confirm.
Puzzle 1 — 1, 2, 6, 24, 120, ?
Not arithmetic (differences 1, 4, 18, 96 — not constant) and not geometric (ratios 2, 3, 4, 5 — not constant either).
The ratios themselves are increasing by 1 each time: ×2, ×3, ×4, ×5 — so the next multiplier is ×6.
Answer: 120 × 6 = 720 (this is actually the sequence of factorials, 1!, 2!, 3!, 4!, 5!, 6!)
Puzzle 2 — 100, 81, 64, 49, 36, ?
Differences: −19, −17, −15, −13 — not constant, so not arithmetic.
Second differences: 2, 2, 2 — constant, so this is a quadratic sequence, descending.
These are perfect squares counting down: 10², 9², 8², 7², 6² — so the answer is 5² = 25
Puzzle 3 — 2, 5, 11, 23, 47, ?
Differences: 3, 6, 12, 24 — these double each time, so the differences themselves are geometric.
The underlying rule is "double the previous term and add 1": aₙ = 2 × aₙ₋₁ + 1.
Answer: 2 × 47 + 1 = 95
Puzzle 4 — 2, 3, 5, 8, 13, ?
Not arithmetic or geometric — but each term is the sum of the two before it, just like Fibonacci, only starting from 2 and 3 instead of 1 and 1.
Compound interest is a geometric sequence — each period multiplies the balance by (1 + rate).
Frequently Asked Questions
An arithmetic sequence has a constant difference (d) between consecutive terms. For 3, 7, 11, 15… the common difference is 4. The nth term formula is aₙ = a₁ + (n−1)d. To find the 10th term: 3 + (10−1)×4 = 39. The sum of n terms is Sₙ = n/2 × (first + last).
A geometric sequence has a constant ratio (r) between consecutive terms. For 2, 6, 18, 54… the ratio is 3. The nth term is aₙ = a₁ × rⁿ⁻¹, and the sum of n terms is Sₙ = a₁(1−rⁿ)/(1−r). Compound interest is a geometric sequence, since each period multiplies the balance by the same (1+rate) factor.
Check the difference between consecutive terms first — if constant, add it to get the next term (arithmetic). If not, check the ratio — if constant, multiply by it (geometric). If neither works, check whether each term is the sum of the previous two (Fibonacci-like), or whether the second differences are constant (quadratic). This calculator runs through all of these checks automatically.
The nth term is a formula that gives any term in a sequence directly from its position n, without needing to list every term before it. For an arithmetic sequence it is aₙ = a₁ + (n−1)d; for a geometric sequence it is aₙ = a₁ × rⁿ⁻¹; for perfect squares it is simply aₙ = n².
The Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13, 21, 34… where each number is the sum of the two preceding numbers. It appears throughout nature — spiral patterns in shells and sunflowers, branching in trees — and the ratio of consecutive terms converges towards the golden ratio, φ ≈ 1.618.
A quadratic sequence is one whose nth term includes an n² component, such as 2, 4, 7, 11, 16 (differences 2, 3, 4, 5; second differences all 1). You spot it by taking the differences between terms, then taking the differences of those differences — if that second layer is constant, the sequence is quadratic, and you can extend it by continuing the second-difference pattern even before finding the full nth-term formula.
Use Sₙ = n/2 × (a₁ + aₙ), where a₁ is the first term, aₙ is the last term, and n is the number of terms. Equivalently, Sₙ = n/2 × (2a₁ + (n−1)d) if you don't already know the last term. For example, the sum of the first 10 terms of 3, 7, 11, … 39 is 10/2 × (3+39) = 210.
For a finite geometric series with ratio r ≠ 1, use Sₙ = a₁(1−rⁿ)/(1−r). For an infinite geometric series where |r| < 1, the terms shrink towards zero and the sum converges to S∞ = a₁/(1−r). If |r| ≥ 1, the infinite sum has no finite value.
Yes. A negative ratio makes the sequence alternate in sign, such as 4, −8, 16, −32 (r = −2). A fractional ratio between −1 and 1 makes the terms shrink towards zero, such as 8, 4, 2, 1, 0.5 (r = 0.5) — this is the case where an infinite geometric sum converges to a finite value.
Three terms are the minimum for detecting arithmetic or geometric patterns, but quadratic pattern detection needs at least four terms, since it relies on comparing multiple second differences. In general, the more known terms you provide, the more confidently the calculator (or a human) can rule out coincidental patterns.
Subtract any term from the term immediately after it: d = aₙ₊₁ − aₙ. For 3, 7, 11, 15, the common difference is 7 − 3 = 4 (or check with any other consecutive pair — 15 − 11 = 4 as well). If the difference isn't the same between every consecutive pair, the sequence isn't arithmetic.
Divide any term by the term immediately before it: r = aₙ₊₁ / aₙ. For 2, 6, 18, 54, the common ratio is 6 ÷ 2 = 3 (check: 54 ÷ 18 = 3 as well). The ratio can be negative (making the sequence alternate in sign) or a fraction between −1 and 1 (making the terms shrink towards zero).
Yes, in everyday and exam usage "sequence" and "progression" are used interchangeably — an "arithmetic progression" (AP) is the same thing as an "arithmetic sequence," and a "geometric progression" (GP) is the same as a "geometric sequence." Some textbooks use "progression" specifically for sequences with a defined algebraic rule (arithmetic, geometric, harmonic) and reserve "sequence" as the more general term that also covers Fibonacci-like or arbitrary patterns, but this distinction is not applied consistently.
Last updated: August 2026
Reviewed by Mohsin Iqbal using standard sequence and series definitions, with every worked example independently verified. This page is for educational purposes.