Higher only · iteration

Iteration.
Feed the answer
back in.

let the ANS button
do the work

the word itself —

Iteration

Latin iterum: again

repeat until the answer settles

step 1 of 4

Show there is a root

1. Substitute both values.

2. The sign changes, so it crosses 0.

✓ There is a root between 1 and 2

x³ − x − 5 = 0

x = 1:  1 − 1 − 5 = −5

x = 2:  8 − 2 − 5 = 1

negative, then positive

step 2 of 4

Read the formula

1. x_n is the current value. x_n+1 is the next.

2. Put 2 in. Out comes the next value.

✓ Write down at least 4 decimal places

x_n+1 = ³√x_n + 5

x₀ = 2 the starting value

x₁ = ³√2 + 5

x₁ = 1.9129…

step 3 of 4

Use the ANS button

1. Type the start value and press =.

2. Type the formula with ANS in place of x.

3. Every press of = gives the next value.

✓ No retyping, no rounding errors

2  = sets ANS to 2

³√ANS + 5  = type once

x₁ = 1.9129   x₂ = 1.9050

x₃ = 1.9042   x₄ = 1.9042

step 4 of 4

When to stop

1. The values stop changing: it has settled.

2. Round to what the question asks for.

✓ That is the root of x³ − x − 5 = 0

1.9129

1.9050

1.9042

1.9042 settled

x = 1.904 (3 d.p.)

iteration —

Sign change

a root lies between

ANS

type the formula once

Settles

that value is the root

Asked for x₃? Count carefully:
the start value is x₀.

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