Identify and generate arithmetic, geometric, Fibonacci, triangular, square, and prime sequences. Find the nth term formula, sum of terms, and next values in the series automatically.
A number sequence is an ordered list of numbers following a specific pattern or rule. Each number in the sequence is called a term. Sequences appear throughout mathematics, science, finance, and nature β from simple counting patterns to complex recursive formulas.
Understanding sequences is fundamental in GCSE and A-level Mathematics in the UK, and in Algebra and AP Calculus in the US. This calculator covers the most common types used in school and professional contexts.
An arithmetic sequence (or arithmetic progression, AP) is a sequence where each term increases or decreases by a constant amount called the common difference d.
Example: 3, 7, 11, 15, ... β a=3, d=4. a(10) = 3 + 9Γ4 = 39. S(10) = 10/2 Γ (6+36) = 5Γ42 = 210.
Arithmetic sequences model linear growth: equal pay raises each year, equal steps in a staircase, equal intervals on a number line. UK GCSE students must find nth term formulas for arithmetic sequences.
A geometric sequence (or geometric progression, GP) is a sequence where each term is multiplied by a constant common ratio r.
Example: 2, 6, 18, 54, ... β a=2, r=3. a(7) = 2Γ3^6 = 2Γ729 = 1,458.
If |r| < 1, the sequence converges to zero and the sum to infinity is Sβ = a/(1βr). This convergence property is the basis of Zeno's paradox and is used in actuarial calculations for perpetuities and annuities.
A savings account earning compound interest is a geometric sequence. If you deposit Β£1,000 annually at 5% interest, each year's ending balance forms a geometric-like series. The present value of a perpetuity (infinite payments) uses the formula PV = C/r, where C is the annual payment and r is the discount rate β directly derived from geometric series convergence.
The Fibonacci sequence is defined recursively: F(1)=1, F(2)=1, F(n)=F(nβ1)+F(nβ2). The sequence begins: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
The ratio of consecutive Fibonacci numbers converges to the golden ratio Ο β 1.6180339887.... This irrational number appears in nature: the spirals of sunflowers, pinecones, and nautilus shells follow Fibonacci counts.
In finance, Fibonacci retracement levels (23.6%, 38.2%, 61.8%) are widely used in technical analysis for identifying potential price reversal levels in stocks and forex.
Triangular numbers: 1, 3, 6, 10, 15, 21, ... where T(n) = n(n+1)/2. They represent the number of dots that can be arranged in an equilateral triangle. T(10) = 55, T(100) = 5,050 (the famous sum Gauss computed as a schoolboy).
Square numbers: 1, 4, 9, 16, 25, ... a(n) = nΒ². Cube numbers: 1, 8, 27, 64, 125, ... a(n) = nΒ³. These appear in geometry (area and volume), physics (inverse square law), and number theory.
Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... A prime has exactly two factors: 1 and itself. There is no general closed-form nth term formula for primes. The prime counting function Ο(x) β x/ln(x) by the Prime Number Theorem. Primes are fundamental to cryptography (RSA encryption, used in every secure website).
| Type | Pattern | nth Term | Example |
|---|---|---|---|
| Arithmetic | +constant | a + (nβ1)d | 2, 5, 8, 11 |
| Geometric | Γconstant | a Γ r^(nβ1) | 3, 6, 12, 24 |
| Fibonacci | Sum of prev two | F(nβ1)+F(nβ2) | 1, 1, 2, 3, 5 |
| Square | nΒ² | nΒ² | 1, 4, 9, 16, 25 |
| Triangular | n(n+1)/2 | n(n+1)/2 | 1, 3, 6, 10, 15 |
| Prime | No simple formula | Sieve of Eratosthenes | 2, 3, 5, 7, 11 |
The AQA, Edexcel, and OCR GCSE Mathematics specifications (Foundation and Higher) require students to: generate terms from a rule, find the nth term of arithmetic and quadratic sequences, and identify whether a given value is in a sequence. Quadratic sequences have second differences that are constant.
The Common Core State Standards (CCSS) include arithmetic and geometric sequences in High School Algebra (HSA-SSE, HSF-BF). AP Calculus BC extends this to convergence and divergence of infinite series (the Ratio Test, Comparison Test, etc.), directly building on geometric sequence concepts.
An arithmetic sequence adds (or subtracts) the same constant to each term. The constant is called the common difference d. The nth term formula is a(n) = a + (nβ1)d, where a is the first term.
A geometric sequence multiplies each term by the same constant ratio r. The nth term is a Γ r^(nβ1). If |r| < 1, terms approach zero and the sum to infinity is a/(1βr).
Formula: a(n) = a + (nβ1)d. Example: for 3, 7, 11, 15 β a=3, d=4. a(n) = 3 + (nβ1)Γ4 = 4nβ1. Check: a(1)=3 β, a(2)=7 β.
Fibonacci: 1,1,2,3,5,8,13,21,... Each term is the sum of the two preceding. It appears in the spiral arrangements of sunflowers (34 and 55 spirals), pinecones, nautilus shells, and the branching of trees.
The golden ratio Ο = (1 + β5)/2 β 1.6180339887. It equals the limit of consecutive Fibonacci ratios (F(n+1)/F(n) β Ο). It appears in art, architecture, and nature, and is considered aesthetically pleasing by many.
S(n) = n/2 Γ (first term + last term) = n/2 Γ (2a + (nβ1)d). For example, sum of first 100 natural numbers = 100/2 Γ (1+100) = 50 Γ 101 = 5,050.
A geometric series converges when |r| < 1. The sum to infinity is Sβ = a/(1βr). For r β₯ 1 or r β€ β1, the series diverges (grows without bound or oscillates without settling).
Triangular numbers T(n) = n(n+1)/2 count handshakes in a group (if n people each shake hands with everyone else), connections in a network, and are related to binomial coefficients C(n+1, 2).
Results are for educational purposes. Auto-detection works best with at least 4 consistent terms. Ambiguous sequences may be misclassified. Verify results against your course specifications.