Iteration (AQA GCSE Maths: Higher): Exam Questions

Show that the equation $x^3+x=7$ has a solution between $1$ and $2$

Show that the equation $x^3+x=7$ can be rearranged to give $x=\sqrt[3]{7-x}$

Starting with $x_0=2,$
use the iteration formula $x_{n+1}=\sqrt[3]{7-x_n}$ three times to find an estimate for a solution of  $x^3+x=7$

Show that the equation $x^3+4x=1$ has a solution between $x=0$ and $x=1$

Show that the equation $x^3+4x=1$ can be arranged to give  $x=\frac{1}{4}-\frac{x^3}{4}$

Starting with $x_0=0$ , use the iteration formula $x_{n+1}=\frac{1}{4}-\frac{x_n^3}{4}$twice , to find an estimate for the solution of  $x^3+4x=1$

Show that the equation $3x^2-x^3+3=0$ can be rearranged to give 

   $x=3+\frac{3}{x^2}$

Using 

      $x_{n+1}=3+\frac{3}{x_n^2}$ with $x_0=3.2,$

find the values of $x_1,x_2$ and $x_3$

Explain what the values of $x_1,x_2,$ and  $x_3$ represent.

A sphere has radius $r$ cm

An approximate value of $r$ can be found using the iterative formula

The starting value is  $r_1=7$

Work out the values of  $r_2$ and $r_3$

$\begin{array}{l}{r}_1=.....................\\ r_2=.....................\end{array}$

Continue the iteration to work out the radius to 1 decimal place.

................................................. cm

$x_{n+1}=\sqrt[3]{3x_n+7}$

Use a starting value of $x_1=2$ to work out a solution to $x=\sqrt[3]{3x+7}$
Give your answer to 3 decimal places.

Using $x_{n+1}=\frac{2}{{x_n}^2}-7$ with $x_0=2$, find the values of $x_1,x_2\mathrm{and}{x}_3$.

Where appropriate, round decimals to 4 decimal places.

Show that the equation $x^3+7x-5=0$ has a solution between $x=0$ and $x=1$

Show that the equation $x^3+7x-5=0$ can be arranged to give $x=\frac{5}{x^2+7}$

Starting with $x_0=1$, use the iteration formula $x_{n+1}=\frac{5}{x_n^2+7}$ three times to find an estimate for the solution of $x^3+7x—5=0$

By substituting your answer to part (c) into $x^3+7x—5,$ comment on the accuracy of your estimate for the solution to $x^3+7x-5=0$

The number of bees in a beehive at the start of year $n$ is $P_n.$ The number of bees in the beehive at the start of the following year is given by 

$P_{n+1}=1.05\left(P_n-250\right)$

At the start of 2015 there were 9500 bees in the beehive.

How many bees will there be in the beehive at the start of 2018?

The number of slugs in a garden $t$days from now is $p_t$ where

   $\begin{array}{l}{p}_0=100\\ p_{t+1}=1.06p_t\end{array}$

Work out the number of slugs in the garden $3$ days from now.

At the start of year $n$, the number of animals in a population is $P_n$
At the start of the following year, the number of animals in the population is $P_{n+1}$, where

$P_{n+1}=k{P}_n$

At the start of 2017 the number of animals in the population was 4000
At the start of 2019 the number of animals in the population was 3610
Find the value of the constant $k$.

Use the iteration formula  $x_{n+1}=\sqrt[3]{10-2x_n}$  to find the values of $x_1$, $x_2$ and $x_3$
Start with $x_0=2$

$x_1=.....$.     
$x_2=.....$.     
$x_3=.....$      

The values of $x_1$, $x_2$ and $x_3$ found part (a) are estimates of the solution of an equation of the form $x^3+ax+b=0$ where $a$ and $b$ are integers.

Find the value of $a$ and the value of $b$.

$a=.....$.     
$b=.....$      

The number of rabbits on a farm at the end of month $n$ is $P_n$ The number of rabbits at the end of the next month is given by $P_{n+1}=1.2P_n-50$

At the end of March there are 200 rabbits on the farm.

Work out how many rabbits there will be on the farm at the end of June.

Considering your results in part (a), suggest what will happen to the number of rabbits on the farm after a long time.