Functions
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Sequences and Series

See number patterns come alive as dots on a graph

Patterns predict the future — from monthly loan payments (arithmetic sequences) to investment growth (geometric sequences) to the spiral of a seashell (Fibonacci). A sequence is a list of numbers that follow a pattern. The simplest kind is an arithmetic sequence like 1, 3, 5, 7, 9, ... where each number is 2 more than the last. Plot these as dots and they form a straight line!

A geometric sequence like 2, 4, 8, 16, 32, ... multiplies by 2 each time. Plot these and the dots curve upward exponentially.

When you ADD up the terms of a sequence, you get a series. Some series grow without bound, but others approach a limit — they converge. The classic example is 1 + 1/2 + 1/4 + 1/8 + ... = 2. In this lesson, you'll see all of this on the graph.

Graph

FAQ

What is a sequence?
A sequence is an ordered list of numbers that follows a rule. Each number is called a term. The first term is a₁, the second is a₂, and so on. For example, 2, 5, 8, 11, ... is a sequence where each term is 3 more than the previous one.
What is the difference between arithmetic and geometric sequences?
An arithmetic sequence adds the same number (the common difference d) each time: a, a+d, a+2d, ... A geometric sequence multiplies by the same number (the common ratio r) each time: a, ar, ar², ... Arithmetic sequences form straight lines when graphed; geometric sequences form exponential curves.
What is a series?
A series is the sum of the terms of a sequence. If you add the first n terms, that's a partial sum Sₙ. For example, the series 1 + 3 + 5 + 7 = 16 is the partial sum of the first 4 terms of the odd numbers sequence. An infinite series adds up infinitely many terms.
What does convergence mean?
A series converges if its partial sums approach a finite number as you add more and more terms. For example, 1 + 1/2 + 1/4 + 1/8 + ... converges to 2. A series diverges if the partial sums grow without bound, like 1 + 2 + 3 + 4 + ... which grows forever.

Related Topics

Linear EquationsExponential Growth & DecayWhat is a Function?