A rational recurrence with period four

A sequence {un}\{u_n\} is defined by

un+1=22−un,n∈Z+,u1≠0,1,2.u_{n+1}=\frac2{2-u_n},\quad n\in\mathbb Z^+,\qquad u_1\ne0,1,2.

(a) Prove that the sequence does not converge. marks

(b) Given u1=4u_1=4: (i) write down the next four terms mark; (ii) prove algebraically that un+4=unu_{n+4}=u_n. marks

[2015/TJC/Promo/6]

Total marks: 7

Solution

(a)

Prove non-convergence

un+1=22−unu_{n+1}=\frac2{2-u_n}, u1≠0,1,2u_1\ne0,1,2. Prove that the sequence does not converge.

Explore this idea

Hint

  1. Assume a finite limit LL and let n→∞n\to\infty in the rule.

  2. Show the resulting equation has no real solution.

Answer · (a)

No finite real limit: a putative limit satisfies (L−1)2+1=0(L-1)^2+1=0; L=2L=2 is a pole.

Explore the idea

The curve never meets the diagonal

Start at the chosen value on the horizontal axis, go up to the curve, across to the diagonal, and repeat. Each corner of the web is the next term.

Starting valueu₁ = 4
Iterations drawn4
Try an example
u(n = 5)
4
u(n = 1)
4
Key idea

A limit must sit where the curve meets the diagonal. Here the curve and the diagonal never meet, which is the picture of (L−1)2+1=0(L-1)^2+1=0.

Ideas to try
  1. Look for an intersection of the teal curve and the red diagonal. There is none, so no value can be a limit.
  2. Sweep the starting value: the web never settles, however many steps you add.
(b)(i)

Compute four terms

Given u1=4u_1=4, write down the next four terms.

Explore this idea

Hint

  1. Apply the rule four times, keeping exact fractions.

Answer · (b)(i)

−1,2/3,3/2,4-1,2/3,3/2,4.

Explore the idea

A closed loop of four corners

Start at the chosen value on the horizontal axis, go up to the curve, across to the diagonal, and repeat. Each corner of the web is the next term.

Starting valueu₁ = 4
Iterations drawn4
Try an example
u(n = 5)
4
u(n = 1)
4
Key idea

A web that closes into a loop is a cycle: the terms repeat in the same order forever.

Ideas to try
  1. Add iterations one at a time from u1=4u_1=4. After four the web closes into a loop and retraces itself.
  2. Move the start to 00, 11 or 22 (the excluded values). The web hits the pole and stops.
(b)(ii)

Prove u(n+4) = u(n)

Prove algebraically that un+4=unu_{n+4}=u_n.

Explore this idea

Hint

  1. Compose the rule with itself to get un+2u_{n+2} in terms of unu_n.

  2. Apply that result twice.

Answer · (b)(ii)

F4(x)=xF^4(x)=x, hence un+4=unu_{n+4}=u_n.

u1≠0,1,2u_1\ne0,1,2 avoids the intervening poles.

Explore the idea

The terms repeat every four steps

The sequence from u1=4u_1=4 cycles through 4, −1, 23, 324,\ -1,\ \frac23,\ \frac32.

Indexn = 5
Try an example
term uₙ at n = 5
4
Key idea

F∘F∘F∘FF\circ F\circ F\circ F is the identity map, so every starting value returns after four steps.

Ideas to try
Compare nn with n+4n+4. The values on the readout are always the same.
Takeaway

A cycle is not a limit

Repetition is not convergence. A sequence that returns to its start every few steps has no limit unless all of its values are equal.