Iteration

The number of bees in a beehive at the start of year  is
The number of bees in the beehive at the start of the following year is given by 

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 days from now is  where

   

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

The number of rabbits on a farm at the end of month  is 
The number of rabbits at the end of the next month is given by 

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 , the number of animals in a population is
At the start of the following year, the number of animals in the population is , where

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 .

Use the iteration formula   to find the values of , and 
Start with

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.     
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The values of , and found part (a) are estimates of the solution of an equation of the form where and are integers.

Find the value of  and the value of .

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.     

Show that the equation  has a solution between  and 

Show that the equation  can be arranged to give  

Starting with , use the iteration formula twice ,
to find an estimate for the solution of  

Show that the equation  can be rearranged to give 

Using 

       with 

find the values of  and 

Explain what the values of and  represent.

A sphere has radius cm

An approximate value of can be found using the iterative formula

The starting value is 

Work out the values of  and

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

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

Use a starting value of to work out a solution to
Give your answer to 3 decimal places.