Geometric sequences

problem
Squaring the circle and circling the square

Age

14 to 16

Challenge level

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

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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?
problem
Converging product

Age

16 to 18

Challenge level

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)... ?
problem
The amazing splitting plant

Age

5 to 7

Challenge level

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

Age

5 to 7

Challenge level

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

Age

7 to 11

Challenge level

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

This article by Alex Goodwin, age 18 of Madras College, St Andrews describes how to find the sum of 1 + 22 + 333 + 4444 + ... to n terms.
interactivity
Proof sorter - geometric sequence

Age

16 to 18

Challenge level

Can you correctly order the steps in the proof of the formula for the sum of the first n terms in a geometric sequence?
problem
Pocket money

Age

11 to 14

Challenge level

Which of these pocket money systems would you rather have?