Iterative formulae, roots and growth models

3 exam-style questions. Answers and the mark for each step are on the last page.

Question 1Calculator · 3 marks
(a) Use the iteration xₙ₊₁ = $\sqrt{}$ (10 + xₙ), starting with x₀ = 3. Work out x₃ to 3 decimal places. (3)
Question 2Calculator · 5 marks
The iteration is $x_{n+1}=\sqrt{10+x_n}$ with $x_0=3$ .
(a) Calculate x4, keeping unrounded values until giving your answer to 3 decimal places. (3)
(b) Write the quadratic equation satisfied by a positive fixed point. (2)
Question 3Calculator · 5 marks
The iteration formula $x_{n+1} = 3 + \dfrac{4}{x_n}$ is used with $x_1 = 1$ .
(a) Work out $x_4$ . Give your answer to 2 decimal places. (2)
(b) The values of $x_n$ get closer to a number $L$ . Show that $L = 4$ . (3)

Answers and marks

(a) $3.700$

(a) $3.701$

M1 Use the previous output as the next input; first compute x1.
M1 The next values are 3.688570357 and 3.699806800; use the third for the fourth update.
A1 Correct answer: $3.701$

(b) $x^{2}-x-10=0$

(a) $4.12$

(b) $L = 3 + \frac{4}{L} \Rightarrow L^2 - 3L - 4 = 0 \Rightarrow L = 4$ (positive)

P1 Writing $L = 3 + \frac{4}{L}$ .
P1 Forming $L^2 - 3L - 4 = 0$ .
C1 Solving to $L = 4$ or $-1$ and rejecting $-1$ because every iterate is positive.

Iterative formulae, roots and growth models

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