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Algebra

Arithmetic Sequence

nth term and sum. Free online Arithmetic Sequence. Calculate arithmetic sequence online — fast, accurate, mobile-friendly, no signup needed.

aₙ
39
Sₙ
210

Derivation

  1. ├── 01Givena1 = 3, d = 4, n = 10
  2. ├── 02Formulaaₙ: t+(e.n-1) × a
  3. ├── 03Substitutet+(e.10-1) × a
  4. ├── 04Compute aₙ39
  5. ├── 05FormulaSₙ: n / 2 × (2 × t+(n-1) × a)
  6. ├── 06Substitute10 / 2 × (2 × t+(10-1) × a)
  7. └── 07Compute Sₙ210
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§01What is

Understanding the Arithmetic Sequence

The Arithmetic Sequence computes aₙ from 3 inputs: first term, common diff, term n. nth term and sum.

Algebra is the art of solving for the unknown. Rearranging a formula to isolate the variable you actually need is the single most common real-world math skill — and doing it with real numbers under time pressure is where errors happen. The Arithmetic Sequence sits in that toolkit — it nth term and sum. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

aₙ = t+(e.n-1) × a | Sₙ = n / 2 × (2 × t+(n-1) × a)

Where

a1
First term
d
Common diff
n
Term n
aₙ
Output value
Sₙ
Output value
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: First term = 3, Common diff = 4, Term n = 10.

  1. 01Start by noting the input — First term: 3.
  2. 02Start by noting the input — Common diff: 4.
  3. 03Start by noting the input — Term n: 10.
  4. 04Substitute these values into the formula: aₙ = t+(e.n-1) × a | Sₙ = n / 2 × (2 × t+(n-1) × a)
  5. 05Compute aₙ: the calculator returns 39.
  6. 06Compute Sₙ: the calculator returns 210.
  7. 07Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Arithmetic Sequence Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

First term halved

a1 = 1.5 (from 3)

Keep every other input at its default and halve the first term. See how aₙ responds.

  1. 01New First term: 1.5
  2. 02Baseline aₙ: 39
  3. 03New aₙ: 37.5
  4. 04aₙ decreases by 3.8% → use this sensitivity to plan for real-world variation.
02 · PATTERN

First term doubled

a1 = 6 (from 3)

Keep every other input at its default and double the first term. See how aₙ responds.

  1. 01New First term: 6
  2. 02Baseline aₙ: 39
  3. 03New aₙ: 42
  4. 04aₙ increases by 7.7% → use this sensitivity to plan for real-world variation.
03 · PATTERN

Common diff halved

d = 2 (from 4)

Keep every other input at its default and halve the common diff. See how aₙ responds.

  1. 01New Common diff: 2
  2. 02Baseline aₙ: 39
  3. 03New aₙ: 21
  4. 04aₙ decreases by 46.2% → use this sensitivity to plan for real-world variation.
04 · PATTERN

Common diff doubled

d = 8 (from 4)

Keep every other input at its default and double the common diff. See how aₙ responds.

  1. 01New Common diff: 8
  2. 02Baseline aₙ: 39
  3. 03New aₙ: 75
  4. 04aₙ increases by 92.3% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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