Take any two positive numbers and call the larger one <i>a</i><sub>1</sub> and smaller <i>b</i><sub>1</sub>. Calculate the arithmetic mean of the two numbers and call this <i>a</i><sub>2</sub>,  where: 





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<i>a</i><sub>2</sub> = (<i>a</i><sub>1</sub>+ <i>b</i><sub>1</sub>)/2.</td></tr></table>
</td></tr></table>




<p/>
Calculate the geometric mean of <i>a</i><sub>1</sub> and <i>b</i><sub>1</sub> and call this <i>b</i><sub>2</sub> so
that: 





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<i>b</i><sub>2</sub> = </td><td nowrap="nowrap" align="center">
</td><td nowrap="nowrap" align="left">&#x00A0;&#x00A0;_____<br /><font size="+1"><font face="symbol">Ö</font></font>(<i>a</i><sub>1</sub><i>b</i><sub>1</sub>)<br />&#x00A0;<br /></td><td nowrap="nowrap" align="center">
.</td></tr></table>
</td></tr></table>




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


<p/>
Repeat the calculations to generate a sequence of arithmetic means <i>a</i><sub>1</sub>,
<i>a</i><sub>2</sub>, <i>a</i><sub>3</sub>, ... and a sequence of geometric means <i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>,
<i>b</i><sub>3</sub>, ... where





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<i>a</i><sub><i>n</i>+1</sub> = (<i>a</i><sub><i>n</i></sub>+ <i>b</i><sub><i>n</i></sub>)/2,</td></tr></table>
</td></tr></table>





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<i>b</i><sub><i>n</i>+1</sub> = </td><td nowrap="nowrap" align="center">
</td><td nowrap="nowrap" align="left">&#x00A0;&#x00A0;_____<br /><font size="+1"><font face="symbol">Ö</font></font>(<i>a</i><sub><i>n</i></sub><i>b</i><sub><i>n</i></sub>)<br />&#x00A0;<br /></td><td nowrap="nowrap" align="center">
.</td></tr></table>
</td></tr></table>




In the examle given 



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<i>a</i><sub>2</sub> = 6.75,</td></tr></table>
</td></tr></table>





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<i>b</i><sub>2</sub> = </td><td nowrap="nowrap" align="center">
</td><td nowrap="nowrap" align="left">&#x00A0;&#x00A0;___<br /><font size="+1"><font face="symbol">Ö</font></font>(45)<br />&#x00A0;<br /></td><td nowrap="nowrap" align="center">
 = 6.708 <i>to</i> 3 <i>decimal</i> <i>places</i>.</td></tr></table>
</td></tr></table>


<p/>
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.


<p/> Now repeat the same calculations starting with different choices of
positive values for <i>a</i><sub>1</sub> and <i>b</i><sub>1</sub>. You should notice the same
behaviour of the two sequences whatever starting values you choose. Describe
and explain this behaviour.


<p/>
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.