When the sequence settles down

7 minA Level3.6.2

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can read and write the notation for a sequence of approximations ( $x_1, x_2, x_3, \dots$ ), work out its terms, and tell when the terms have settled down to a root.

Guess, improve, repeat

Many equations, like $x^3 - 2x - 5 = 0$ , have no neat algebraic way to solve them. The previous lesson used a sign change to show roughly where a root is. This lesson is about closing in on it.

The plan:

Make a first guess.
Feed it into a formula that produces a better guess.
Feed that back into the same formula. Repeat.

Each pass is called an iteration. If the formula is a good one, the guesses get closer and closer to the root.

The notation

$x_1$ is the first guess (the starting value). The small number underneath is just a counter: $x_1$ is "guess number 1".
$x_2, x_3, x_4, \dots$ are the next guesses, in order.
$x_n$ means "guess number $n$ ", any one of them, and $x_{n+1}$ is the guess straight after it.
The formula is written $x_{n+1} = F(x_n)$ : "the next guess is $F$ of the current guess". So $x_2 = F(x_1)$ , $x_3 = F(x_2)$ , and so on.
The exact root is usually called $\alpha$ (the Greek letter "alpha").

If the guesses get as close to $\alpha$ as you like, the sequence converges to $\alpha$ .

Worked example: Example 1: an iteration for √2

The iteration $x_{n+1} = \dfrac12\left(x_n + \dfrac{2}{x_n}\right)$ with $x_1 = 1$ converges to $\sqrt2$ . Find $x_2$ , $x_3$ and $x_4$ .

Have a go first?

Why this formula works

$\tfrac12\left(x_n + \tfrac{2}{x_n}\right)$ is the average of $x_n$ and $\tfrac{2}{x_n}$ .

If $x_n$ is a bit too small, then $\tfrac{2}{x_n}$ is a bit too big, because their product is always exactly 2, just like $\sqrt2 \times \sqrt2$ . So one is below $\sqrt2$ and the other is above. Their average lands in between. For a reasonable guess it lands much closer to $\sqrt2$ than either of them, which is why the errors shrink so fast.

You don't need to invent formulas like this; exam questions give them to you. You just need to use them correctly.

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 2 questions above to continue.