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Sequences and Series Explained

A sequence is an ordered list of numbers that follow a rule; a series is the sum of the terms of a sequence. Whether you are calculating how long it takes to pay off a loan, modelling population growth, or working out how much a ball travels before coming to rest, arithmetic and geometric sequences and their series formulas are the tools for the job.

Key Terminology

Each number in a sequence is called a term. The first term is usually written a₁ (or simply a), the second a₂, and the general term (the nth term) is a_n or u_n. A sequence can be defined by an explicit formula giving a_n in terms of n, or by a recurrence relation giving each term in terms of the previous one (e.g. a_n+1 = a_n + 3).

Arithmetic Sequences

In an arithmetic sequence, each term is obtained from the previous one by adding a fixed amount called the common difference d. The sequence 3, 7, 11, 15, 19, ... has a = 3 and d = 4. The nth term formula is:

a_n = a + (n − 1)d

For example, the 20th term of 3, 7, 11, 15, ... is 3 + 19 × 4 = 3 + 76 = 79. If d is positive the sequence increases; if negative it decreases. A zero common difference produces a constant sequence.

The sum of the first n terms of an arithmetic sequence is called an arithmetic series. There are two equivalent formulas; use whichever is easier given the information provided:

S_n = (n/2)(a + l)    (where l is the last term)

S_n = (n/2)(2a + (n − 1)d)

Legend has it that the young Gauss astonished his teacher by instantly summing 1 + 2 + 3 + ... + 100 = (100/2)(1 + 100) = 50 × 101 = 5050, using the first formula intuitively.

Geometric Sequences

In a geometric sequence, each term is obtained from the previous one by multiplying by a fixed number called the common ratio r. The sequence 2, 6, 18, 54, ... has a = 2 and r = 3. The nth term formula is:

a_n = a × rⁿ⁻¹

Notice that the index is n − 1, not n, because the first term (n = 1) is just a (no multiplications yet). If r > 1 the sequence grows without bound. If 0 < r < 1 the terms decrease toward zero. If r is negative, terms alternate in sign. If r = −1, the sequence alternates between +a and −a.

Arithmetic vs Geometric: Side-by-Side Comparison

Sum of a Geometric Series

The sum of the first n terms of a geometric series is:

S_n = a(1 − rⁿ) / (1 − r)    (for r ≠ 1)

An equivalent form, often easier when r > 1, is S_n = a(rⁿ − 1) / (r − 1). For r = 1 all terms are equal and S_n = na.

Example: Find the sum of the first 8 terms of 3, 6, 12, 24, ... (a = 3, r = 2).

S₈ = 3(2⁸ − 1) / (2 − 1) = 3 × 255 / 1 = 765.

Infinite Geometric Series and Convergence

If the common ratio satisfies |r| < 1, successive terms get closer and closer to zero, and the infinite series converges to a finite sum:

S_∞ = a / (1 − r)    (valid only when |r| < 1)

For example, consider the series 1 + 1/2 + 1/4 + 1/8 + ... (a = 1, r = 1/2). S_∞ = 1 / (1 − 1/2) = 1 / (1/2) = 2. This is the mathematical resolution of Zeno’s paradox: a runner who covers half the remaining distance with each step does eventually cross the finish line, because the sum of an infinite geometric series with |r| < 1 is finite. If |r| ≥ 1, the terms do not approach zero and the infinite series diverges — its sum grows without bound.

Real-World Applications

Arithmetic sequences appear in fixed-rate savings (earning the same interest amount each year), salary scales with equal annual increments, and the triangular numbers (1, 3, 6, 10, ...). Geometric sequences appear in compound interest and exponential growth, radioactive decay (each half-life halves the remaining atoms), and the bouncing of a ball (each bounce reaching a fixed fraction of the previous height). The sum of a bouncing ball’s total distances before it stops is an infinite geometric series — convergent because each bounce is shorter than the last.

Summary

Arithmetic sequences add a constant difference d; their nth term is a + (n−1)d and partial sum is (n/2)(2a + (n−1)d). Geometric sequences multiply by a constant ratio r; their nth term is arⁿ⁻¹ and partial sum is a(1−rⁿ)/(1−r). Infinite geometric series converge to a/(1−r) only when |r| < 1. These tools connect to many areas of mathematics, from probability to financial modelling, and are foundational for calculus and Taylor series at higher levels.