Sequence Calculator
Arithmetic and geometric sequences: the nth term, the sum of the first n terms, the first ten terms listed, and the infinite sum when the series converges.
Sequence Calculator: with the default inputs, nth term is 48.
How much is added to get from one term to the next.
What each term is multiplied by. |r| < 1 makes the infinite sum converge.
- Sum of the first n terms
- 255
- First ten terms
- 3, 8, 13, 18, 23, 28, 33, 38, 43, 48
- Sum to infinity
- —Geometric sequences with |r| < 1 only — otherwise the series diverges.
- Behaviour
- Increasing by 5 each step
Assumptions
- Terms are indexed from n = 1, so the first term is a₁.
- The geometric sum uses the closed form a₁(1 − rⁿ)/(1 − r), with the r = 1 case handled separately.
- The infinite sum is reported only when |r| < 1; otherwise it is left blank rather than being computed from a formula that does not apply.
| n | Term | Running total |
|---|---|---|
| 1 | 3 | 3 |
| 2 | 8 | 11 |
| 3 | 13 | 24 |
| 4 | 18 | 42 |
| 5 | 23 | 65 |
| 6 | 28 | 93 |
| 7 | 33 | 126 |
| 8 | 38 | 164 |
| 9 | 43 | 207 |
| 10 | 48 | 255 |
The running total in the last row of the full sequence is the sum of the first n terms.
How this is worked out
The formula
Arithmetic (add d each time): aₙ = a₁ + (n − 1)d Sₙ = (n ÷ 2)(a₁ + aₙ) = (n ÷ 2)(2a₁ + (n − 1)d) Geometric (multiply by r each time): aₙ = a₁ · r^(n − 1) Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for r ≠ 1; Sₙ = a₁n when r = 1 S∞ = a₁ ÷ (1 − r) only when |r| < 1
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Sequence type
- Choose one of 2 options.Arithmetic — add the same amount each time · Geometric — multiply by the same amount each time
- First term (a₁)
- A number.defaults to 3
- Common difference (d)
- How much is added to get from one term to the next.defaults to 5
- Common ratio (r)
- What each term is multiplied by. |r| < 1 makes the infinite sum converge.defaults to 2
- Which term? (n)
- A number.from 1 to 100000 · whole numbers only · defaults to 10
What you get back
- nth termmain answer
- Sum of the first n terms
- First ten terms
- Sum to infinity
- Geometric sequences with |r| < 1 only — otherwise the series diverges.
- Behaviour
What this assumes
- Terms are indexed from n = 1, so the first term is a₁.
- The geometric sum uses the closed form a₁(1 − rⁿ)/(1 − r), with the r = 1 case handled separately.
- The infinite sum is reported only when |r| < 1; otherwise it is left blank rather than being computed from a formula that does not apply.
About this calculator
Two patterns cover most of the sequences you will ever meet. An arithmetic sequence adds the same amount every step (3, 8, 13, 18…). A geometric sequence multiplies by the same amount every step (2, 6, 18, 54…). This calculator gives the nth term, the running total, and — for geometric sequences that shrink — the sum all the way to infinity.
How to use it
Pick the type, give the first term, and give either the common difference d (arithmetic) or the common ratio r (geometric). Set n to the term you want. The table lists the terms with a running total, so you can check the closed-form sum against the plain addition.
Where the sum formula comes from
The arithmetic sum has the neatest derivation in mathematics, and the story attached to it is probably true. Write the sequence forwards, write it again backwards underneath, and add the columns: every column gives the same total, a₁ + aₙ. There are n columns and you counted everything twice, so Sₙ = (n ÷ 2)(a₁ + aₙ). That is how a young Gauss reportedly added 1 to 100 in seconds: 50 pairs of 101 = 5,050.
For a geometric sum the trick is different. Multiply Sₙ by r, subtract it from Sₙ, and almost everything cancels, leaving a₁ − a₁rⁿ = Sₙ(1 − r).
The infinite sum, and when it exists
If |r| < 1 the terms shrink towards zero fast enough that the total settles on a finite number: S∞ = a₁ ÷ (1 − r). Half plus a quarter plus an eighth plus … really does add to exactly 1. That is Zeno's paradox resolved in one line.
If |r| ≥ 1 the terms do not shrink, and there is no sum to infinity — the calculator says so rather than printing a number. The 1 + 2 + 4 + 8 + … = −1 you may have seen quoted is a different kind of summation (analytic continuation, or 2-adic arithmetic), not this one.
Where people go wrong
- Off-by-one on n. The nth term uses (n − 1), not n, because the first term already counts as one step taken. The 10th term of 3, 8, 13… is 3 + 9×5 = 48, not 53.
- Confusing aₙ with Sₙ. One is a single term, the other the total so far.
- Using the infinite formula when |r| ≥ 1. It will happily produce a number — a negative one, often — that means nothing.
- Compound growth is geometric. A balance growing 5% a year is a geometric sequence with r = 1.05, and its "sum" is the total of the yearly balances, not the final balance. That distinction trips people up constantly.
Frequently asked questions
▸What is the formula for the nth term of an arithmetic sequence?
aₙ = a₁ + (n − 1)d. Subtract one from n because the first term is already in place before any difference is added.
▸How do I find the sum of an arithmetic sequence?
Sₙ = (n ÷ 2)(a₁ + aₙ) — the number of terms times the average of the first and last. It works because pairing terms from the two ends always gives the same total.
▸When does an infinite geometric series have a sum?
Only when |r| < 1. Then S∞ = a₁ ÷ (1 − r). If |r| ≥ 1 the terms never shrink to nothing and the series diverges.
▸What is the difference between a sequence and a series?
A sequence is the ordered list of terms; a series is what you get when you add them up. This calculator gives both.
▸Is compound interest a geometric sequence?
Yes — each year's balance is the previous one times (1 + rate), so r = 1.05 for 5% growth. Just remember that the balance is a term, not the sum of the sequence.
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