Iteration (Edexcel 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 $r_3$

$$\begin{gathered} r_1=..................... \\ {r}_2=..................... \end{gathered}$$

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.

At the start of the year $n$, the number of trees in a forest is $T_n$.

At the start of the following year, the number of trees is given by

$T_{n+1}=k{T}_n$

where $k$ is a positive constant.

The number of trees at the start of the first year is $10000$.

The number of trees at the start of the second year is $8000$.

Work out the number of trees at the start of the third year.

At the start of the fifth year, the value of $k$ is increased by 0.25 to a new constant value.

Juwariah thinks that from the start of the fifth year, the number of trees in the forest will increase each year.

Is Juwariah correct?

You must give a reason for your answer.

The value of a second-hand watch at the start of the year $n$ is $V_n$.

The value of the watch at the start of the year $n+1$ is given by the formula $V_{n+1}=r{V}_n$

Given that $r$ has a constant value of 0.85, find how many years it takes for the value of the watch to fall to half of its original value.

You must show your working clearly.

Sofia says the value of the watch may start to increase again in the future.

Give one reason why this might happen.

Write the next 3 iterations using $x_{n+1}=12-5x_n$ with $x_1=3$.

Show that the equation $x^3-4x+2=0$ can be rearranged to give

$x=\sqrt[3]{4x-2}$

Using

$x_{n+1}=\sqrt[3]{4x_n-2}$ with $x_0=1$,

find the values of $x_1,x_2\mathrm{and}{x}_3$

Explain what the values of $x_1,x_2\mathrm{and}{x}_3$ represent.

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

   $$\begin{gathered} p_0=100 \\ {p}_{t+1}=1.06p_t \end{gathered}$$

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

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.

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 $b$.

$a=.....$.     
$b=.....$.     

A wildlife ranger is studying the population of a rare species of frog on a small island.

The number of frogs on the island at the start of the year $n$ is $P_n$
The ranger models the change in population each year using the formula

$P_{n+1}=0.82P_n+12$

At the start of the first year of the study, the ranger records the population of frogs as 40.

The study is planned to last for 3 years.

How many frogs will there be on the island at the end of the study?