Sequence & Series Calculator (AP, GP & Pattern Finder)
Arithmetic, geometric and harmonic progressions, pattern finder and exact Σ / Π.
Result
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Terms
About the Sequence & Series Calculator (AP, GP & Pattern Finder)
For an arithmetic or geometric progression, type whatever you know — the first term, the common difference or ratio, the number of terms, the last term, the sum or the sum to infinity — and the calculator finds the rest, with the formula and every substitution shown. It finds n even when that needs a quadratic (sometimes two values of n work), tells you whether a number is a term at all, and inserts arithmetic, geometric or harmonic means between two numbers.
The pattern finder works out the rule behind a list such as 2, 5, 8, 11 or 1, 1, 2, 3, 5, 8 — finite differences for polynomial rules, common ratios, Fibonacci-type recurrences, interleaved patterns, primes and other named sequences — and gives the next terms. The Σ / Π mode adds up or multiplies terms exactly (fractions stay fractions), gives closed forms such as Σ k² = n(n + 1)(2n + 1)/6, and sums convergent geometric series. All arithmetic is exact and runs in your browser.
How to use it
- Choose AP, GP, HP, Find the pattern, or Σ / Π.
- For a progression, fill in the values you know (at least three, or a and d / a and r) and leave the others empty — or type the first few terms. Open “Insert means” to put numbers between two given ones.
- For a pattern, type at least three terms separated by commas. For Σ or Π, type the term (such as k^2), the index and the limits; the upper limit can be a number, a letter such as n, or ∞.
- Press Calculate (results also update as you type). Read the values, the terms, the working and the notes; copy or download them as text.
Examples
a = 5, d = 3, n = 20
a₂₀ = 62, S₂₀ = 670
a₂₀ = 5 + 19·3 = 62 and S₂₀ = 20/2 × (5 + 62) = 670.
AP 24, 21, 18, … with Sₙ = 78
n = 4 or n = 13
−3n² + 51n − 156 = 0 has both roots; the terms from the 5th to the 13th add up to 0.
Is 301 a term of 5, 11, 17, …?
No
n = (301 − 5)/6 + 1 = 151/3 is not a whole number.
GP 9, 6, 4, …
S∞ = 27
r = 2/3, so S∞ = a/(1 − r) = 9/(1/3) = 27.
GP 0.3, 0.03, 0.003, …
S∞ = 1/3
So 0.333… = 1/3.
Insert 6 means between 3 and 24
6, 9, 12, 15, 18, 21
1, 2, 4, 7, 11, 16
aₙ = (n² − n + 2)/2; next 22, 29, 37
The differences 1, 2, 3, 4, 5 go up by 1, so the second differences are constant.
Σ k² for k = 1 to n
n(n + 1)(2n + 1)/6
Common uses
- Solving Class 10 arithmetic progression questions: n-th term, sum of n terms, how many terms, is it a term.
- Class 11 sequences and series: geometric progressions, sums to infinity, arithmetic and geometric means.
- Working out the rule and the next numbers of a number pattern in a puzzle or aptitude test.
- Checking sigma-notation sums and products, with exact fractions and closed forms.
The formulas used
- Arithmetic progression (NCERT Class 10, Arithmetic Progressions): aₙ = a + (n − 1)d and Sₙ = (n/2)[2a + (n − 1)d] = (n/2)(a + aₙ).
- Geometric progression (NCERT Class 11, Sequences and Series): aₙ = a·rⁿ⁻¹, Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1 (Sₙ = n·a when r = 1), and S∞ = a/(1 − r) when |r| < 1.
- Means: inserting k arithmetic means between A and B gives d = (B − A)/(k + 1); k geometric means give r = (B/A)^(1/(k + 1)); harmonic means are the reciprocals of the arithmetic means of 1/A and 1/B. For two positive numbers, AM ≥ GM ≥ HM and GM² = AM × HM.
- Harmonic progression: the reciprocals form an AP, so aₙ = 1/(1/a + (n − 1)d) with d = 1/b − 1/a. There is no simple formula for its sum, so the terms are added exactly.
How the pattern finder decides
It tries rules from the simplest: a constant, an AP, a GP, prime numbers, a Fibonacci-type rule (each term the sum of the two before), named sequences (squares, cubes, triangular numbers, factorials, powers of 2, Catalan numbers), polynomial rules found by finite differences (when the k-th differences are constant the n-th term is a polynomial of degree k), differences or ratios that themselves form a pattern, other recurrences from the two previous terms, and two patterns interleaved in odd and even positions. Every rule is checked against all the terms you typed. Any finite list of numbers fits infinitely many rules, so the answer is the simplest of these that fits — a sensible guess, not a proof.
Σ and Π
With numeric limits every term is computed exactly and the results are added (Σ) or multiplied (Π); if a term is irrational, such as √k, the result is a decimal instead. With a letter as the upper limit, a polynomial term gives a polynomial closed form (Faulhaber’s formulas, found exactly from the first few partial sums), for example Σ k³ = n²(n + 1)²/4, and a geometric term c·rᵏ gives c(rⁿ⁺¹ − rᵃ)/(r − 1). Infinite sums are given exactly for geometric series with |r| < 1 and for Σ 1/k², 1/k⁴ and 1/k⁶ (π²/6, π⁴/90, π⁶/945); the harmonic series Σ 1/k diverges.
Limitations
- Inputs must be rational numbers (whole numbers, decimals or fractions). A common ratio found from aₙ can be a surd such as ∛2, but ratios found from a sum are searched for among rational numbers only.
- Finite Σ and Π take at most 100,000 terms. The result is exact while the fraction stays manageable; when it grows too large to finish within a few seconds (Σ 1/k beyond a few thousand terms), the rest is added in double precision and the result is given as a decimal. Exact results of more than 60,000 digits are stopped.
- Closed forms with a letter as the upper limit are given for polynomial and geometric terms only; other infinite series are not summed.
Privacy
Everything happens in your browser. What you enter or open here is not uploaded or stored by MySmartCoPilot.
Frequently asked questions
How do I find the number of terms of an AP?
If you know a, d and the last term, n = (aₙ − a)/d + 1; it must come out as a positive whole number, otherwise that number is not a term. If you know the sum instead, Sₙ = (n/2)[2a + (n − 1)d] is a quadratic in n — and when d is negative it can have two positive whole-number solutions, both correct.
When does a geometric series have a sum to infinity?
Only when the common ratio is between −1 and 1 (|r| < 1, r ≠ 0). Then the terms shrink towards 0 and the sum approaches S∞ = a/(1 − r). For 9 + 6 + 4 + …, r = 2/3 and S∞ = 27. An arithmetic series never has a finite sum to infinity unless every term is 0.
What is the difference between a sequence and a series?
A sequence is a list of terms (3, 7, 11, 15, …); a series is the sum of those terms (3 + 7 + 11 + 15 + …). Sₙ is the sum of the first n terms.
How does the calculator find the next number in a pattern?
It looks for a rule that produces every term you typed — constant differences, a constant ratio, a sum of the two previous terms, a polynomial found by repeated differences, and so on — and applies it to the next positions. Because many rules can fit a short list, give as many terms as you can.
What is the sum of the first n squares?
Σ k² for k = 1 to n is n(n + 1)(2n + 1)/6; for n = 10 that is 385. Choose Σ / Π, type k^2, from 1, to n for the formula or to 10 for the number.
How do I insert geometric means between two numbers?
To put k numbers between A and B so that all of them form a GP, use r = (B/A)^(1/(k + 1)). Between 1 and 256 with three means, r⁴ = 256, so r = 4 and the means are 4, 16, 64 (with r = −4 you get −4, 16, −64).