GCSE · Maths · AQA · Spec 8300 · Higher

General iterative processes (Higher)

Feed a rule its own answer, over and over, and one simple rule can predict a fish population years ahead.

Maths · Iteration

One rule, fed its own answers

The rule: multiply by 3, then subtract 1. The start: x₀ = 4. In symbols, xₙ₊₁ = 3xₙ − 1.

Follow one number through the rule. Watch where each answer goes next.

Uses	In	Working	Out
0			x₀ = 4

Output

x₀ = 4

Step 1: We start at x₀ = 4. The little 0 means the rule has been used zero times — nothing has happened to it yet.

1 / 4

Press Next to use the rule once more. Each press is one more iteration.

Step 1 of 4: We start at x₀ = 4. The little 0 means the rule has been used zero times — nothing has happened to it yet..

Watch out: Easy mistake: putting 4 back in every time. That just gives 11, then 11 again, then 11 again — you never get anywhere.

Iteration in context

Predicting a fish population

A lake holds 14,000 fish. Each year 2,000 fish are removed, and then the fish that remain grow in number by 18%. So p₀ = 14,000 and pₙ₊₁ = 1.18(pₙ − 2,000). Estimate the population after two years.

p₀ = 14,000 — the population now, before any years have passed.Subscript 0: no years yet.
missing step

Which line is step 2?

p₁ = 1.18 × (14,000 − 2,000)p₁ = 1.18 × 14,000 − 2,000p₁ = 0.18 × (14,000 − 2,000)

Iterating on a calculator with Ans

By hand is great for seeing the idea. For speed, let the Ans key carry each answer into the next step for you.

Type the start and press =Type the starting value and press =. The calculator stores it as Ans. The rule hasn't been used yet, so the screen is showing x₀.
Press = onceThe calculator drops Ans into the rule and shows the result: x₁. One press, one use of the rule — and Ans quietly updates to this new value.
Keep pressing =Every press feeds the latest answer straight back in. After n presses you're looking at xₙ, so counting presses tells you exactly which value is on the screen.
Write each value downThe screen only ever shows the newest value, so jot each one down as it appears, rounded the way the question asks — whole fish for a population.

What does the answer mean?

Reading a model's value

The fish model starts at p₀ = 14,000 and gives p₁ = 14,160 and p₂ = 14,349.

What does p₂ = 14,349 tell you? Pick the idea closest to what you think right now.

It's the population after one year — p₁ is the start, so p₂ is the first new value.It's the model's estimate of how many fish there will be after two years.It's exactly how many fish will be in the lake in two years' time.It's how many extra fish the lake has gained over the two years.

How sure are you?

I'm sureI think soI'm guessing

It's the population after one year — p₁ is the start, so p₂ is the first new value.

This is a common misconception

This is the off-by-one slip. The start is p₀, not p₁. p₁ is after one year, so p₂ is after two. The subscript counts how many times the rule has been used.

It's the model's estimate of how many fish there will be after two years.

This matches the evidence

Exactly. The subscript counts years here — p₀ now, p₁ after one year, p₂ after two — and the model gives an estimate, not a guarantee.

It's exactly how many fish will be in the lake in two years' time.

This is a common misconception

The model assumes exactly 2,000 are removed and exactly 18% growth every year, and it rounds to whole fish. A real lake won't follow those numbers perfectly, so 14,349 is an estimate.

It's how many extra fish the lake has gained over the two years.

This is a common misconception

p₂ is the whole population, not the change. The gain over two years is 14,349 − 14,000 = 349 fish.

A value like p₂ is the whole amount after that many steps of the model — and it's an estimate, not a promise.

When to stop

Stop, keep going, or give up?

You're iterating to find a value to a required accuracy. Where has each one got to?

Still to sort

Stop: the answer is reached (0)

Several outputs in a row agree at the accuracy you need.

Where the line is: One matching output isn't enough. You stop only when several successive outputs round to the same value.

Keep iterating (0)

Not settled yet, but nothing has gone wrong.

Iteration has failed (0)

The next step can't be done, or the values will never settle.

6 of 6 still to sort.

Sometimes you iterate to home in on a value at a given accuracy. Decide where each iteration has got to.

Watch out: If an iteration fails, don't write the formula off straight away. A different starting value may lead to a different outcome.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Feed a rule its own answer, again and again: whatever comes out goes back in, and one rule builds a whole run of values.

What you need to know

An iterative process applies the same rule again and again, and the output of each step becomes the input for the next.
The starting value has subscript zero (x₀, p₀); xₙ is the value after the rule has been used n times.
In a context such as a population model, do the steps in the order the words give, and round as the context demands — whole numbers for a population.
To reach an approximate value, iterate until several successive outputs round to the same value. Iteration can fail: a square root of a negative number, a division by zero, or values that never settle.

The big picture

Iteration means feeding a rule its own answer: whatever comes out goes straight back in as the next input, so one rule produces a whole run of values. The starting value gets subscript zero, like x₀, and xₙ is the value after the rule has been used n times. The same idea can model real situations, such as a fish population year by year, where you follow the order of the steps and round to whole fish. When you're homing in on a value, you stop once several outputs in a row agree — and you watch for the ways an iteration can fail.

Key points

1Out goes in: the previous output is always the next input — never the starting value again.

2x₀ is the start, x₁ comes after one use of the rule, x₃ after three.

3On a calculator, Ans holds the last answer, so each press of = is one more iteration.

4One matching output isn't enough: stop when several in a row agree at the accuracy you need.

5If an iteration fails, a different starting value may lead to a different outcome.

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Worked example

Problem

The fish model (remove 2,000 fish, then grow by 18%) gives p₂ = 14,349. Estimate the population after three years, and say which value of p this is.

⚠ Watch out

Putting the starting value back in at every step. Each new step uses the previous output — so for the fish, year 2 starts from 14,160, not 14,000.

🧠

Memory hook

Out goes in. Think of a machine you feed its own output: whatever comes out, you drop straight back in the top — and the subscript is a tally of how many times you've turned the handle.

✓

Check yourself

Starting from x₀ = 4 with the rule “multiply by 3, then subtract 1”, what is x₄ — and how many times have you used the rule to get it?

Flashcards

(13)

What is an iterative process?

Applying the same rule again and again, where the output of each step becomes the input for the next.

What does the subscript 0 in x₀ or p₀ tell you?

It's the starting value: the rule has been used zero times.

What is x₁?

The value after the rule has been used once on x₀.

What does xₙ mean?

The value after the rule has been used n times from the start.

How do you work out the next value from an iterative formula?

Substitute the previous output — not the starting value — into the rule.

In the fish model, which comes first: removing 2,000 or growing by 18%?

Remove the 2,000 first, then multiply what's left by 1.18.

Why multiply by 1.18 for an 18% increase?

1 keeps everything you already have, and 0.18 adds the 18% on top — both in one multiplication.

How are values in a population model rounded?

To the nearest whole number — you can't have part of an animal.

How does the Ans key speed up iteration?

Ans holds the last answer, so after typing the rule once, each press of = applies it again without retyping.

When do you stop iterating to reach an approximate value?

When several successive outputs round to the same value at the required accuracy.

Name three ways an iteration can fail.

A step needs the square root of a negative number; a step divides by zero; the values never settle on a limit.

An iteration fails from one starting value. What might you try?

A different starting value — it may lead to a different outcome.

What does a model value like p₂ tell you?

The model's estimate of the whole amount after two steps (for example, two years) — not an exact count.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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One rule, fed its own answers

Predicting a fish population

Reading a model's value

Stop, keep going, or give up?

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