Every sequence hides a rule. Type in a few numbers below, or click through the tabs to build, break, and predict sequences, arithmetic & geometric progressions, Gauss's summing trick, and the fractals they create.
Pick a sequence family and watch its dot pattern grow. Triangular numbers stack like bowling pins; square numbers form perfect squares.
An explicit rule finds the nth term directly from its position. A recursive rule builds each term from the one(s) before it. Try both below.
An AP adds a fixed common difference d each step. Its stage-vs-value graph is always a straight line — that's the signature of "linear growth."
Write the sum forwards and backwards, pair them up — every pair adds to the same total! Slide n and watch the pairing.
A GP multiplies by a fixed common ratio r each step. Its graph curves sharply — that's exponential growth (or decay if r < 1).
Each bounce reaches a fixed fraction r of the previous height. Press play and watch the GP in motion.
Remove the middle triangle, repeat on what's left. Black-triangle count follows tₙ=3ⁿ (growing GP); black area follows sₙ=(3/4)ⁿ (shrinking GP)!
Trisect each side, remove the centre square, repeat on the 8 remaining squares. Red-square count follows tₙ=8ⁿ; red area follows sₙ=(8/9)ⁿ.
Six quick questions covering everything in this lab. Click an option to see if you're right.