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.

Related Topics

Linear EquationsExponential Growth & DecayWhat is a Function?

Suggested Lessons

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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.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.