Sequences & Series
A sequence is a list of numbers with a rule; a series is what you get when you add them up. Watch both at once — the terms as bars, the running total as a line — and the formulas stop being things to memorise. For Grade 11.
Two rules, two families
- Arithmetic — each term adds a fixed difference d: 2, 5, 8, 11, … The terms grow in a straight line.
- Geometric — each term multiplies by a fixed ratio r: 4, 2, 1, ½, … The terms curve.
Build the sequence 📊
Switch between Arithmetic and Geometric, then slide the first term and the difference or ratio. Blue bars are the terms tₙ; green dots are the running total Sₙ (the series).
What to notice
The sum is the running total
Each green dot is every blue bar so far, added up. In arithmetic the dots curve upward faster and faster — that's why the sum formula Sₙ = n2(2a + (n−1)d) has an n² in it.
When a geometric series settles
Switch to Geometric and set the ratio between −1 and 1. The terms shrink to nothing, so the running total stops climbing and homes in on a limit — the dashed line S∞ = a1 − r. Push the ratio past 1 and it never settles.
Try it: set the ratio to 0.6 and watch the green dots creep toward the dashed limit but never quite reach it — an infinite sum with a finite answer. Then nudge the ratio to 1.5 and the whole thing takes off. That single threshold at |r| = 1 is the heart of the topic, and here you can see it flip.