Arithmetic & Geometric Sequence Calculator – Tutorial
On this page, you can find the logic, usage, and important details of the Arithmetic & Geometric Sequence Calculator calculator.
What Is a Sequence?
A sequence is an ordered list of numbers following a specific rule: a₁, a₂, a₃, …, aₙ. Two fundamental types stand out: arithmetic and geometric. They form the backbone of school mathematics and underlie countless models in finance, engineering and science.
Arithmetic Sequence
In an arithmetic sequence the difference between consecutive terms is constant. This constant is called the common difference (d):
n-th Term Formula
Starting at a₁ and adding d each step:
a₃ = a₁ + 2d
aₙ = a₁ + (n − 1) · d
Sum Formula — Gauss Pairing Trick
To find Sₙ = a₁ + a₂ + … + aₙ, write the sequence forwards and backwards. Each column pair sums to (a₁ + aₙ), and there are n such pairs:
Sₙ = n · (a₁ + aₙ) / 2
Example
a₅ = 3 + (5−1)·4 = 19
S₅ = 5·(3 + 19)/2 = 55
Where Is It Used?
- Regular saving, running or production goals that increase by the same amount each period
- Linear growth models, evenly spaced measurements
Geometric Sequence
In a geometric sequence the ratio between consecutive terms is constant. This constant is called the common ratio (r):
n-th Term Formula
a₃ = a₁ · r²
aₙ = a₁ · rⁿ⁻¹
Sum Formula
Multiplying Sₙ = a₁ + a₁r + … + a₁rⁿ⁻¹ by r and subtracting gives:
Sₙ = n · a₁ [r = 1]
Example
a₅ = 2 · 3⁴ = 162
S₅ = 2 · (1 − 3⁵) / (1 − 3) = 2 · (−242) / (−2) = 242
Where Is It Used?
- Finance: Compound interest, investment growth (fixed percentage each period)
- Physics & biology: Radioactive half-life, bacterial growth, damped oscillations
- Computer science: Algorithmic scaling, logarithmic growth
Arithmetic or Geometric?
| Check | Result | Type |
|---|---|---|
| Consecutive difference constant? | Yes → d fixed | Arithmetic |
| Consecutive ratio constant? | Yes → r fixed | Geometric |
Example: 3, 7, 11, 15 → difference 4 (arithmetic). 2, 6, 18, 54 → ratio 3 (geometric).
Things to Watch Out For
- n must be a positive integer; n = 1 means the sequence has one term and the sum equals a₁.
- In a geometric sequence, r = 0 makes all terms from a₂ onward equal to zero.
- For |r| > 1 and large n, values grow extremely fast and numeric overflow (Infinity) may occur.
- The table shows at most 15 terms; the Sₙ column is the running (cumulative) sum up to that term.
Note: Results are for educational use. Floating-point rounding may affect precision for large n or r.