 # Exam-Style Questions on Iteration

## Problems on Iteration adapted from questions set in previous Mathematics exams.

### 1.

GCSE Higher

Using $$x_{n+1}=-5-\frac{6}{x_n^2}$$

with $$x_0 = -1.5$$

(a) find the values of $$x_1, x_2$$ and $$x_3$$

(b) Explain the relationship between the values of $$x_1, x_2$$ and $$x_3$$ and the equation $$x^3+5x^2+6=0$$

### 2.

GCSE Higher

(a) A sequence is defined by the following rule where $$u_n$$ is the $$n^{th}$$ term of the sequence:

$$u_{n+1}=u_n^2-5u_n+21$$

If $$u_1=3$$ find $$u_2$$ and $$u_3$$.

(b) A different sequence is defined by the following rule:

$$u_{n+1}=u_n^2-8u_n+17$$

If $$u_1=5$$ find $$u_2$$ and $$u_3$$ and $$u_{50}$$.

### 3.

GCSE Higher

Consider the following cubic equation:

$$x^3-7x-5=0$$

An approximate solution can be found by using the following iterative process.

$$x_{n+1}=\frac{(x_n)^3-5}{7}$$

(a) Find $$x_2$$ and $$x_3$$ if $$x_1=-1$$

Work out the solution to 6 decimal places.

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