Recursive Sequences

Calculus 2 · Axiom Academy

Each term is built from the one before it — and a cobweb plot shows where the whole chain is headed. A recursive sequence is defined by two things: a starting value a_0 , and a rule a_ n+1 =f(a_n) that produces the next term from the current one. You generate the sequence by feeding each output back in as the next input. A starting value, then one rule applied over and over Each arrow is one application of f 2. Fixed Points and Equilibria A fixed point (or equilibrium) is a value L that the rule sends to itself: f(L)=L . Once the sequence lands on L , it never moves again. Geometrically, L is exactly where the curve y=f(x) crosses the diagonal y=x . Stable ( ): nearby sequences are pulled toward L . Unstable ( ): nearby sequences are pushed away from L . f(x)=0.5x+1 gives , and f'(x)=0.5 everywhere. Since , the fixed point L=2 is stable — which is why closes in on 2 . A cobweb diagram traces the iteration on a single graph. Draw both y=f(x) and the diagonal y=x , then bounce between them: the curve turns the current term into the next, and the diagonal carries that output back down to become the next input. Start at (a_0, 0) ; go vertically to the curve at . Go horizontally to the diagonal at , since f(a_0)=a_1 . Go vertically to the curve again at . Repeat: vertical to the curve, horizontal to the diagonal. For a linear rule f(x)=mx+b , the slope m at the fixed point decides everything. Three regimes appear — each is just a different cobweb shape:

This is the written version of the interactive lesson above. See the full Calculus 2 course.