Sequences and series are the building blocks of quantitative thinking. An arithmetic sequence adds a fixed step each time; a geometric sequence multiplies by a fixed ratio. Understanding the difference — and knowing how to compute any term or partial sum instantly — is useful everywhere from personal finance to physics.
Arithmetic vs. Geometric: Which Pattern Do You Have?
The fastest way to classify a sequence is to check the differences between consecutive terms. If the differences are all equal (e.g. 3, 7, 11, 15 — difference of 4 each time), it's arithmetic. If the ratios are all equal (e.g. 2, 6, 18, 54 — ratio of 3 each time), it's geometric. The calculator's Custom mode auto-detects these patterns for you.
Real-world arithmetic sequences include annual salary raises, straight-line depreciation, and evenly spaced appointments. Geometric sequences appear in compound interest, population growth, radioactive decay, and signal attenuation — anywhere a percentage change repeats each period.
When Does a Geometric Series Converge?
A geometric series converges — meaning its partial sums settle toward a single finite number — if and only if the absolute value of the common ratio is less than 1 (|r| < 1). The limiting value is S∞ = a₁ / (1 − r). For example, the series 1 + ½ + ¼ + ⅛ + … (r = 0.5) converges to exactly 2.
When |r| ≥ 1 the terms either stay constant (r = 1), alternate in sign without shrinking (r = −1), or grow without bound (|r| > 1). In all three cases the series diverges — there is no finite sum to infinity. The calculator flags this with a Diverges badge and hides the S∞ field.
The Fibonacci Sequence and the Golden Ratio
Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21 …) are neither arithmetic nor geometric, yet they appear throughout nature — in sunflower seed spirals, nautilus shells, and branching trees. The ratio of consecutive Fibonacci terms (e.g. 21/13 ≈ 1.615) converges to the golden ratio φ = (1 + √5) / 2 ≈ 1.618 as n → ∞.
Practical uses include algorithm analysis (Fibonacci heaps, recursive Fibonacci complexity) and financial trading (Fibonacci retracement levels). This calculator shows any Fibonacci term up to F₇₈ — beyond that JavaScript's floating-point precision cannot represent the integer exactly.