Take any two positive numbers and call the larger one

a_{1}

and smaller

b_{1}

. Calculate the arithmetic mean of the two numbers and call this

a_{2}

, where:

a_{2} = (a_{1} + b_{1}) / 2 .

Calculate the geometric mean of $a_{1}$ and $b_{1}$ and call this $b_{2}$ so that:

$b_{2} = \sqrt{(a_{1} b_{1})} .$

Suppose you start with 3 and 12, then the arithmmmetic mean is 7.5 and the geometric mean is 6.

Repeat the calculations to generate a sequence of arithmetic means $a_{1}$ , $a_{2}$ , $a_{3}$ , ... and a sequence of geometric means $b_{1}$ , $b_{2}$ , $b_{3}$ , ... where

$a_{n + 1} = (a_{n} + b_{n}) / 2,$

$b_{n + 1} = \sqrt{(a_{n} b_{n})} .$

In the examle given

$a_{2} = 6.75,$

$b_{2} = \sqrt{(45)} = 6.708 to 3 decimal places .$

Calculate the first 5 terms of each sequence and mark them on a number line. Calculate a few more terms and make a note of what happens to the two sequences.

Now repeat the same calculations starting with different choices of positive values for $a_{1}$ and $b_{1}$ . You should notice the same behaviour of the two sequences whatever starting values you choose. Describe and explain this behaviour.

You may like to write a short program for a calculator or computer to calculate the sequences and if so you should send in your program with your solution.