What is the sum of: 6 + 66 + 666 + 6666 ............+ 666666666...6 where there are n sixes in the last term?

Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

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?

How can visual patterns be used to prove sums of series?

Which of these pocket money systems would you rather have?

Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?

What is the sum of: 6 + 66 + 666 + 6666 ............+ 666666666...6 where there are n sixes in the last term?

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

Make a poster using equilateral triangles with sides 27, 9, 3 and 1 units assembled as stage 3 of the Von Koch fractal. Investigate areas & lengths when you repeat a process infinitely often.

What is the total area of the triangles remaining in the nth stage of constructing a Sierpinski Triangle? Work out the dimension of this fractal.

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

Can you predict, without drawing, what the perimeter of the next shape in this pattern will be if we continue drawing them in the same way?

The interval 0 - 1 is marked into halves, quarters, eighths ... etc. Vertical lines are drawn at these points, heights depending on positions. What happens as this process goes on indefinitely?

These grids are filled according to some rules - can you complete them?

Can you correctly order the steps in the proof of the formula for the sum of a geometric 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.

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)?

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