This sequence calculator handles the two great sequence families: arithmetic (add the same difference each step) and geometric (multiply by the same ratio) — nth term, sum of the first n terms, and the first ten terms printed for a sanity check.
Two Families, Two Futures
Arithmetic sequences walk; geometric sequences compound. a₁ = 3, d = 4 reaches 39 by term 10; a₁ = 2, r = 3 reaches 486 by term 6 and 39,366 by term 10 — the same "growth," different physics. Anything genuinely compounding (money at interest, populations early on) belongs to the geometric family, whose continuous cousin is our Exponential Growth Calculator. Part of our Math Calculators collection.
Worked Examples (Verified)
- Arithmetic a₁=3, d=4, n=10 (defaults): a₁₀ = 3 + 9×4 = 39; S₁₀ = 10/2 × (3 + 39) = 210
- Geometric a₁=2, r=3, n=6: a₆ = 2×3⁵ = 486; S₆ = 2×(729−1)/2 = 728
- The Gauss anecdote, checkable here: a₁=1, d=1, n=100 → S = 5,050 — the schoolboy's pairing trick (100 pairs summing 101, halved) is the Sₙ formula in disguise
- Decay belongs too: r = 0.5, a₁ = 100 → terms 100, 50, 25, 12.5… — half-life mathematics wearing a sequence's clothes
Reading a Sequence from Its Terms
- Constant differences between neighbors → arithmetic; constant ratios → geometric. Check both before modeling.
- Neither constant? Look one level down (second differences constant → quadratic sequence) or at ratios of differences — the Fibonacci family hides there, with its own ratio converging to φ.
- Sums have closed forms only for friendly families — arithmetic and geometric are the two every curriculum (and every loan) is built on.
Related Tools
- Exponential Growth Calculator — geometric growth, continuously
- Factorial Calculator — growth that outruns both families
- Math Calculators — the full collection
Frequently Asked Questions
What is the formula for the nth term of an arithmetic sequence?
aₙ = a₁ + (n − 1)d. For a₁ = 3, d = 4: the 10th term is 3 + 9×4 = 39. The (n−1) appears because the first term needs zero steps of the difference added.
How do you sum a geometric series?
Sₙ = a₁(rⁿ − 1)/(r − 1) for r ≠ 1. With a₁ = 2, r = 3, n = 6: 2 × 728/2 = 728. When |r| < 1 and n → ∞, the infinite sum converges to a₁/(1 − r) — 100, 50, 25… sums to exactly 200.
How can I tell if a sequence is arithmetic or geometric?
Subtract consecutive terms: constant result → arithmetic. Divide them: constant result → geometric. Some sequences are neither (squares: differences 3, 5, 7…) and a few degenerate ones are both (constant sequences).
Where do sequences show up in real life?
Arithmetic: fixed deposits, seat rows, simple interest. Geometric: compound interest (each term × (1 + rate)), depreciation (× (1 − rate)), viral spread, radioactive decay chains. Choosing the wrong family is the most common spreadsheet forecasting error — the growth shapes diverge fast.
What is the sum of 1 to 100?
5,050 — enter a₁ = 1, d = 1, n = 100 in arithmetic mode. The formula Sₙ = n(a₁ + aₙ)/2 gives 100 × 101/2; Gauss allegedly produced it in seconds as a schoolboy by pairing 1+100, 2+99…
Is this sequence calculator free?
Yes — free, instant, no sign-up.