Arithmetic & Geometric Sequences
A sequence is an ordered list of numbers — 2, 5, 8, 11, … — but it is far more powerful to think of a sequence as a function. Specifically, a sequence is a function whose domain is the set of positive integers (or sometimes non-negative integers). Instead of writing f(x) for a continuous function, we write a(n) or simply aₙ, where n = 1, 2, 3, … tells us the position, and a(n) gives the value at that position. This functional perspective connects sequences directly to the broader world of functions you study in algebra and calculus. Two families of sequences appear everywhere in mathematics, science, and finance: arithmetic sequences and geometric sequences. An arithmetic sequence changes by adding the same constant each time (like your savings growing by $50 every month), while a geometric sequence changes by multiplying by the same constant each time (like a bacterial population doubling every hour). Each family has its own explicit formula — a direct formula giving a(n) in terms of n — as well as a recursive formula that defines each term using the previous one. Understanding sequences as functions also unlocks exponential and logarithmic connections. The explicit formula for a geometric sequence is literally an exponential function restricted to integer inputs, which is why this topic lives inside the unit on exponential and logarithmic functions. Mastering both types of sequences prepares you for series, limits, compound interest, and real-world modeling tasks throughout higher mathematics.
- Sequences as Functions: The Domain and Notation
- Arithmetic Sequences: Add a Constant
- Geometric Sequences: Multiply by a Constant
- Recursive vs. Explicit Formulas
- Identifying Which Type You Have
Sequences as Functions: The Domain and Notation
A sequence a: ℕ → ℝ assigns a real number to each positive integer n. We write a(n) or aₙ. The first term is a(1) = a₁, the second is a(2) = a₂, and so on. Thinking this way, every sequence is just a function with a discrete (whole-number) domain, and its graph consists of isolated dots at (1, a₁), (2, a₂), (3, a₃), …
Arithmetic Sequences: Add a Constant
An arithmetic sequence has a first term a₁ and a common difference d. Each term is obtained by adding d to the previous term: a(n) = a(n−1) + d. The explicit (closed-form) formula is a(n) = a₁ + (n − 1)d. This is a linear function of n — its graph is a set of collinear points with slope d. Example: if a₁ = 3 and d = 4, then a(n) = 3 + (n − 1)·4 = 4n − 1, giving 3, 7, 11, 15, …
Geometric Sequences: Multiply by a Constant
A geometric sequence has a first term a₁ ≠ 0 and a common ratio r ≠ 0. Each term is r times the previous term: a(n) = r · a(n−1). The explicit formula is a(n) = a₁ · rⁿ⁻¹. This is an exponential function of n — its graph curves upward (if r > 1), decays toward zero (if 0 < r < 1), or alternates sign (if r < 0). Example: if a₁ = 5 and r = 2, then a(n) = 5 · 2ⁿ⁻¹, giving 5, 10, 20, 40, …
Recursive vs. Explicit Formulas
A recursive formula defines a(n) in terms of previous terms plus a base case. It is easy to write but requires computing every earlier term to find a later one. An explicit formula defines a(n) directly from n, so you can jump straight to, say, the 100th term without listing the first 99. Both forms are useful: recursive formulas appear naturally in computer algorithms and difference equations, while explicit formulas are better for analysis and graphing.
Identifying Which Type You Have
Given a sequence, check whether consecutive differences are constant (arithmetic) or whether consecutive ratios are constant (geometric). For any three consecutive terms a, b, c: if b − a = c − b, the sequence is arithmetic with d = b − a. If b/a = c/b (and a ≠ 0), the sequence is geometric with r = b/a. Some sequences are neither — for example, 1, 4, 9, 16, … (perfect squares) is neither arithmetic nor geometric.
The Key Equation
- a(n) — nth term (same units as a₁): The output of the sequence function at position n; the value of the sequence's nth element.
- a_1 — first term (same units as a(n)): The starting value of the sequence, found when n = 1.
- r — common ratio (dimensionless): The constant multiplier between consecutive terms. If |r| > 1 the sequence grows; if 0 < |r| < 1 it decays toward zero.
- n — term index (positive integer): The position of the term in the sequence. The domain of the sequence function is n = 1, 2, 3, …
- n-1 — exponent (dimensionless): Because the first term uses zero multiplications by r, the exponent is one less than the term number.
Worked Example
Problem. A geometric sequence has first term a₁ = 6 and common ratio r = 3. (a) Write the explicit formula. (b) Find the 7th term. (c) Which term first exceeds 4000?
Answer. Explicit formula: a(n) = 6·3ⁿ⁻¹; 7th term = 4374; the 7th term is the first to exceed 4000.
Origins · 1 minLeonhard Euler and how this was discovered
Euler systematized much of the notation and theory used for sequences and series. He introduced the Σ (sigma) notation for sums, studied infinite geometric series rigorously, and connected exponential functions to complex numbers through his famous formula e^(iπ) + 1 = 0. His textbook Introductio in Analysin Infinitorum (1748) treated sequences and their generating functions in ways that still influence how we teach the subject today. Euler lost the sight in his right eye in his late 20s and became almost completely blind by age 59, yet he produced roughly half of his total mathematical output after losing his sight — dictating proofs and calculating entirely in his head.
Related Concepts
- Geometric Series and Their Sums — A geometric series is the sum of all terms of a geometric sequence. The closed-form sum formula S = a₁(1 − rⁿ)/(1 − r) follows directly from the explicit sequence formula and is essential for finance, probability, and calculus.
- Exponential Functions — The explicit formula for a geometric sequence, a(n) = a₁·rⁿ⁻¹, is an exponential function evaluated at integer inputs. Understanding continuous exponential functions f(x) = abˣ deepens your grasp of why geometric sequences grow or decay the way they do.
- Logarithmic Functions — Logarithms are the tool for solving geometric-sequence questions like 'which term first exceeds a given value?' — as demonstrated in the worked example above, where log₃ is used to isolate n in an exponential inequality.
- Linear Functions — An arithmetic sequence a(n) = a₁ + (n−1)d is a linear function of n with slope d. Recognizing this link lets you use slope-intercept reasoning to quickly find formulas and interpret arithmetic sequences graphically.