A square-root iteration crosses its limit

A sequence of positive numbers satisfies

xn+1=10−3xn(n=1,2,3,…).x_{n+1}=\sqrt{10-3x_n}\quad(n=1,2,3,\ldots).

(i) Given convergence to ll, find ll. marks

(ii) Prove xn+12−l2=3(l−xn)x_{n+1}^2-l^2=3(l-x_n). marks

(iii) Use (ii) to show that if xn<lx_n<l, then xn+1>lx_{n+1}>l. marks

[2012/IJC/Prelim/II/3]

Total marks: 6

Solution

(i)

Fixed point

A positive sequence satisfies xn+1=10−3xnx_{n+1}=\sqrt{10-3x_n}. Given that it converges to ll, find ll.

Explore this idea

Hint

  1. Replace both xnx_n and xn+1x_{n+1} by ll.

  2. Square, solve the quadratic and reject any negative root.

Answer · (i)

l=2l=2.

Explore the idea

The only positive fixed point

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 = 1
Iterations drawn6
Try an example
u(n = 7)
1.82513
u(n = 1)
1
Distance to l = 2
0.17487
Key idea

The negative root −5-5 is not a candidate: the curve is a positive square root and never reaches the negative axis.

Ideas to try
Step through the iterations. The web spirals in to the point where the curve meets the diagonal.
(ii)

Prove an identity

Prove that xn+12−l2=3(l−xn)x_{n+1}^2-l^2=3(l-x_n).

Explore this idea

Hint

  1. Square the recurrence to remove the root.

  2. Substitute l=2l=2.

Answer · (ii)

xn+12−l2=3(l−xn)x_{n+1}^2-l^2=3(l-x_n).

Explore the idea

Two sides of the identity coincide

For x1=1x_1=1, the quantity xn+12−4x_{n+1}^2-4 and the quantity 3(2−xn)3(2-x_n) agree at every nn.

Indexn = 2
Try an example
x²ₙ₊₁ − l² at n = 2
-1.9373
3(l − xₙ) at n = 2
-1.9373
Key idea

The identity links the sign of the next gap xn+12−l2x_{n+1}^2-l^2 to the sign of the present gap l−xnl-x_n.

Ideas to try
Compare the two readouts at each nn. They match, and both change sign each step.
(iii)

Sign argument

Use (ii) to show that if xn<lx_n<l, then xn+1>lx_{n+1}>l.

Explore this idea

Hint

  1. Read the sign of 3(l−xn)3(l-x_n).

  2. Factor the left side as a difference of squares.

Answer · (iii)

xn<2⇒xn+1>2x_n<2\Rightarrow x_{n+1}>2.

For the prescribed positive, defined sequence.

Explore the idea

Each step jumps across the limit

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 = 1
Iterations drawn3
Try an example
u(n = 4)
2.38565
u(n = 1)
1
Distance to l = 2
0.38565
Key idea

A decreasing curve passing through the fixed point maps the left of it to the right, so the web always crosses.

Ideas to try
  1. From a start below 22, the first step lands above 22. From a start above 22, it lands below.
  2. Open the terms plot: the points alternate around the guide line at 22.
Takeaway

Sign of the gap

Writing the recurrence's effect on xn−lx_n-l in a factored form lets you read off whether the next term overshoots.