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Iteration
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Foundational skills
Identify starting value in iteration
\[ x_0 = 2.5 \rightarrow \text{start} \]
Identify what x₀ represents.
Identify terms in iteration sequence
\[ x_3 = \text{?} \]
Read values from a sequence.
Performing iterations
Substitute into square root formula
\[ x_{n+1} = \sqrt{ax_n + b} \]
Perform one iteration with √.
Substitute into cube root formula
\[ x_{n+1} = \sqrt[3]{a \pm bx_n} \]
Perform one iteration with ∛.
Substitute into fraction formula
\[ x_{n+1} = \frac{a + x_n^2}{b} \]
Perform one iteration with fractions.
Perform three or more iterations
\[ x_0 \rightarrow x_3 \text{ or } x_4 \]
Find x₃ or x₄ from x₀.
Find missing value in sequence
\[ x_0, x_1, \text{?}, x_3 \]
Find a missing term.
Accuracy and convergence
Round iteration results
\[ 2.3456 \rightarrow 2.35 \]
Round to specified d.p.
Identify when iterations converge
\[ x_5 \approx x_6 \text{ to ? d.p.} \]
Find convergence accuracy.
State the solution from results
\[ x_6 = 2.345 \rightarrow 2.35 \]
State final solution.
Rearranging for iteration
Rearrange quadratic to iterative form
\[ x^2 + bx + c = 0 \rightarrow x = … \]
Rearrange for iteration.
Rearrange cubic to iterative form
\[ x^3 + bx – c = 0 \rightarrow x = … \]
Rearrange cubic for iteration.
Write iterative formula from rearrangement
\[ x = \sqrt{…} \rightarrow x_{n+1} = … \]
Convert to iterative form.
Write iterative formula from cubic
\[ x = \sqrt[3]{…} \rightarrow x_{n+1} = … \]
Convert cubic to iterative form.
Connecting formulas and equations
Identify equation from √ formula
\[ x_{n+1} = \sqrt{…} \rightarrow x^2 + … = 0 \]
Find quadratic from formula.
Identify equation from ∛ formula
\[ x_{n+1} = \sqrt[3]{…} \rightarrow x^3 + … = 0 \]
Find cubic from formula.
Solving equations
Solve equation using iteration
\[ x_{n+1} = \sqrt{…}, x_0 = 2 \]
Use formula to find solution.
Solve quadratic using iteration
\[ x^2 + bx + c = 0 \]
Complete solution process.
Solve cubic using iteration
\[ x^3 + bx + c = 0 \]
Complete cubic solution.
Extensions
Determine accuracy from iterations
\[ x_5, x_6 \rightarrow \text{? d.p. accurate} \]
Find solution accuracy.
Identify which root iteration approaches
\[ \text{roots } 2, 5 \rightarrow \text{which?} \]
Determine convergence target.
Timer (Optional)
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