Sequence formulas and symbols
Indexing starts at 1: a1 is the first term, an is the term at position n, d is the common difference, r is the common ratio, and Sn is the sum of the first n terms.
Arithmetic term rules
Explicit gives any term directly; recursive adds d to the previous term.
Arithmetic finite sum
Sₙ = n(a₁ + aₙ) / 2
Sₙ = n[2a₁ + (n − 1)d] / 2
These equivalent forms use the average of the endpoints or substitute the explicit nth term.
Geometric term rules
aₙ = a₁rn−1
aₙ = r · aₙ₋₁
Explicit uses a power of r; recursive multiplies the previous term by r.
Geometric sums
Sₙ = a₁(1 − rn) / (1 − r)
S∞ = a₁ / (1 − r), |r| < 1
For r = 1, use Sn = na1. An infinite sum exists only when |r| < 1.
How to identify the sequence type
Find a difference
From adjacent terms, d = ak+1 − ak. From indexed terms, d = (an − am)/(n − m), then a1 = am − (m − 1)d.
Find a ratio
For adjacent terms with a nonzero preceding term, r = ak+1/ak. Constant ratios identify geometric sequences; division by zero is undefined.
Negative and special values
A negative difference can decrease. A negative ratio alternates signs. Ratio 0 makes every term after the first zero; ratio 1 makes a constant sequence.
Sequence versus series
A sequence is an ordered list of terms. A series is their sum. A pasted list may be arithmetic, geometric, both when constant, or neither.
Worked examples
Arithmetic: a1 = 3, d = 4, n = 6
- Terms: 3, 7, 11, 15, 19, 23.
- Nth term:
a₆ = 3 + (6 − 1)·4 = 23. - Finite sum:
S₆ = 6(3 + 23)/2 = 78.
Answer: a6 = 23 and S6 = 78.
Geometric fraction: a1 = 8, r = 1/2, n = 5
- Terms: 8, 4, 2, 1, 1/2.
- Nth term:
a₅ = 8(1/2)⁴ = 1/2. - Finite sum:
S₅ = 8[1 − (1/2)⁵]/(1 − 1/2) = 31/2 = 15.5. - Because |1/2| < 1,
S∞ = 8/(1 − 1/2) = 16.
Answer: a5 = 1/2, S5 = 31/2, and S∞ = 16.
Arithmetic from two indexed terms: a3 = 10 and a7 = 26
d = (26 − 10)/(7 − 3) = 4, then a₁ = 10 − (3 − 1)·4 = 2. Therefore aₙ = 2 + (n − 1)·4.
Math review, privacy, and limits
Method reviewed: formulas and edge cases were checked against OpenStax Algebra and Trigonometry, Key Equations. No individual credentialed reviewer is attributed because one is not currently available.
Exact integer and fractional arithmetic is retained whenever practical. Irrational roots and logarithmic position solutions use decimal approximations. Inputs are processed locally, n is limited to 100,000, and the display is capped at 50 terms. Extremely large exact powers and non-finite decimal results are rejected clearly.
| Case | Input | Expected | Status |
|---|---|---|---|
| Positive arithmetic | a₁=3, d=4, n=10 | a₁₀=39, S₁₀=210 | Pass |
| Negative difference | a₁=20, d=−3, n=8 | a₈=−1, S₈=76 | Pass |
| Zero ratio | a₁=5, r=0, n=4 | a₄=0, S₄=5 | Pass |
| Fractional ratio | a₁=8, r=1/2, n=5 | a₅=1/2, S₅=31/2 | Pass |
| Ratio 1 | a₁=7, r=1, n=6 | a₆=7, S₆=42 | Pass |
| Convergent infinite | a₁=8, r=1/2 | S∞=16 | Pass |
| Overflow guard | a₁=10¹⁰⁰, r=10¹⁰⁰, n=100000 | Clear size error | Pass |
Sequence calculator FAQs
What is the difference between an arithmetic and geometric sequence?
An arithmetic sequence adds the same difference each time. A geometric sequence multiplies by the same ratio each time.
How do I find the common difference?
Subtract an adjacent term from the next term: d = aₖ₊₁ − aₖ. From two indexed terms, use d = (aₙ − aₘ)/(n − m).
How do I find the common ratio?
Divide an adjacent nonzero term by the term before it: r = aₖ₊₁/aₖ. If the preceding term is zero, that division is undefined.
How do I solve a sequence from two known terms?
Use “From two indexed terms.” Arithmetic mode finds d and then a₁. Geometric mode solves rⁿ⁻ᵐ = aₙ/aₘ and then finds a₁, when a real ratio is identifiable.
When does an infinite geometric series converge?
Only when |r| < 1. Then S∞ = a₁/(1 − r). If |r| ≥ 1, the infinite series diverges.
Can I enter fractions?
Yes. Use integers, decimals, or fractions such as 1/2, -3/4, and 2/3. Exact fractional answers appear first wherever practical, with a decimal approximation second.
Why do r = 0 and r = 1 need special handling?
With r = 0, every term after the first is zero and some inverse problems have multiple solutions. With r = 1, every term equals a₁, so the finite sum is na₁ instead of the usual fraction with denominator 1 − r.
What is the difference between explicit and recursive formulas?
An explicit formula calculates aₙ directly from n. A recursive formula starts with a₁ and uses the preceding term: aₙ = aₙ₋₁ + d or aₙ = r·aₙ₋₁.
Can a pasted sequence be neither arithmetic nor geometric?
Yes. If neither the consecutive differences nor the valid consecutive ratios stay constant, the calculator identifies the list as neither.
Why are only 50 terms available in the table?
The safety cap keeps the page responsive. Paging changes which of those rows are visible; nth-term and finite-sum calculations still support n up to 100,000.