The short answer
Arithmetic sequences add a constant amount each step (2, 5, 8, 11... adds 3 every time); geometric sequences multiply by a constant ratio (3, 6, 12, 24... doubles every time). Once you know which type a sequence is and its starting value, a formula gives you any term — the 100th, the 1000th — without listing everything in between.
Key takeaways
- Arithmetic sequences have a constant difference between consecutive terms; geometric sequences have a constant ratio.
- The nth-term formulas let you jump straight to any position — aₙ = a₁ + (n−1)d for arithmetic, aₙ = a₁ × r^(n−1) for geometric — without generating every term first.
- Not every sequence is arithmetic or geometric — Fibonacci-style sequences build each term from the previous ones instead of a constant difference or ratio.
- A sum formula exists for both sequence types, so you can total the first n terms without adding them one by one.
Recognizing the pattern type
| Pattern | Test | Example |
|---|---|---|
| Arithmetic | Consecutive differences are constant | 2, 5, 8, 11 (d=3) |
| Geometric | Consecutive ratios are constant | 3, 6, 12, 24 (r=2) |
| Fibonacci-style | Each term = sum of previous terms | 1, 1, 2, 3, 5, 8 |
The nth-term and sum formulas
| Type | nth term | Sum of first n terms |
|---|---|---|
| Arithmetic | aₙ = a₁ + (n−1)d | Sₙ = n/2 × (a₁+aₙ) |
| Geometric | aₙ = a₁ × r^(n−1) | Sₙ = a₁(rⁿ−1)/(r−1) |
Worked example: sum of an arithmetic sequence
Find the sum of the first 10 terms of 2, 5, 8, 11... (a₁ = 2, d = 3).
a₁₀ = 2 + (10−1) × 3 = 29
S₁₀ = 10/2 × (2 + 29) = 5 × 31 = 155
The sum formula only needs the first term, the last term, and the count — no need to add all 10 numbers by hand to check the total.
Common mistakes to avoid
- Assuming a pattern is arithmetic after checking just one pair of terms — confirm every consecutive difference matches before concluding.
- Using the arithmetic sum formula on a geometric sequence, or vice versa — the two formulas aren't interchangeable.
- Forgetting the geometric sum formula breaks down when r = 1 — every term is identical, so the sum is simply n × a₁, not a division by zero.
- Treating a recursive sequence like Fibonacci as if it had a simple closed-form nth-term formula — it doesn't, without more advanced identities.
Related calculators
- Average Calculator — compare the mean of your sequence's terms against its formula-based sum.
- Exponent Calculator — check the powers behind a geometric sequence's growth.
- Ratio Calculator — verify the common ratio driving a geometric sequence.
- Statistics Calculator — analyze a generated sequence as a full data set.