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:
  $$b_2 = \sqrt{(a_1b_1)}.$$ Suppose you start with 3 and 12, then
  the arithmmmetic mean is 7.5 and the geometric mean is 6. <br />
  <br />
  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 $$a_{n+1} = (a_n+ b_n)/2,$$
  $$b_{n+1} = \sqrt{(a_nb_n)}.$$In the examle given $$a_2 = 6.75,$$
  $$b_2 = \sqrt{(45)}= 6.708\; \mbox{to 3 decimal
  places}.$$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. <br />
  <br />
  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. <br />
  <br />
  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. <br />