### Bat Wings

Three students had collected some data on the wingspan of some bats. Unfortunately, each student had lost one measurement. Can you find the missing information?

### A Mean Tetrahedron

Can you number the vertices, edges and faces of a tetrahedron so that the number on each edge is the mean of the numbers on the adjacent vertices and the mean of the numbers on the adjacent faces?

### Pairs

Ann thought of 5 numbers and told Bob all the sums that could be made by adding the numbers in pairs. The list of sums is 6, 7, 8, 8, 9, 9, 10,10, 11, 12. Help Bob to find out which numbers Ann was thinking of.

# Converging Means

##### Stage: 3 Challenge Level:
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.

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.

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.