Sequence Detective

Intro to Proofs · Axiom Academy

Cracking a sequence by eye feels like enough — until the pattern lies to you. The fix is a method you can prove. The case of the sequence that lies Spotting a pattern in a few numbers feels like solving the case. But a sequence can match your guess perfectly for a while and then break — and "it looked obvious" is not a proof. Watch one sequence walk you straight into the trap. Mark some points around a circle, connect every pair with a chord, and count the regions the circle is cut into. With 1,2,3,4,5 points you get 1, 2, 4, 8, 16 — powers of two, screaming "next is 32 ." Watch what the sixth point actually does. The pattern was never "double it." Five terms agreed by coincidence — and coincidence is exactly what a proof would have caught. Drag the slider to add points around the circle. Every new point connects to all the others, and the regions are counted from the picture in front of you — no formula assumed. Compare the true count to what "keep doubling" predicts, and watch them split apart. The honest count comes from — every crossing of two chords, plus every chord, plus the whole disk. It only impersonates for the first five terms. Build a sequence you can prove Instead of guessing the next term, state a rule that pins it down: each term is the sum of the two before it. That is a recurrence relation . Slide to grow the Fibonacci sequence and watch every term get built — never guessed — from F(n)=F(n-1)+F(n-2) .

This is the written version of the interactive lesson above. See the full Intro to Proofs course.