If a number N is expressed in binary by using only 'ones,' what can
you say about its square (in binary)?
What is the sum of: 6 + 66 + 666 + 6666 ............+ 666666666...6 where there are n sixes in the last term?
Cellular is an animation that helps you make geometric sequences
composed of square cells.
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
What is the sum of: 6 + 66 + 666 + 6666 ............+ 666666666...6
where there are n sixes in the last term?
This is an interactivity in which you have to sort into the correct
order the steps in the proof of the formula for the sum of a
When is a Fibonacci sequence also a geometric sequence? When the
ratio of successive terms is the golden ratio!
Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?
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)... ?
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
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.
Generalise the sum of a GP by using derivatives to make the
coefficients into powers of the natural numbers.
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?
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?
Each week a company produces X units and sells p per cent of its
stock. How should the company plan its warehouse space?
What is the total area of the triangles remaining in the nth stage
of constructing a Sierpinski Triangle? Work out the dimension of
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.