Geometric sequences

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

Can you work out how many flowers there will be on the Amazing Splitting Plant after it has been growing for six weeks?

On Friday the magic plant was only 2 centimetres tall. Every day it doubled its height. How tall was it on Monday?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

Which of these pocket money systems would you rather have?