Geometric sequences

problem
Favourite

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
Favourite

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

Transformations Tables

Age

7 to 11

Challenge level

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

Mobile Numbers

Age

5 to 11

Challenge level

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?
problem

Clickety Click

Age

16 to 18

Challenge level

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

Generally Geometric

Age

16 to 18

Challenge level

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

Golden Fibs

Age

16 to 18

Challenge level

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

Sierpinski Triangle

Age

16 to 18

Challenge level

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.
problem

Production Equation

Age

16 to 18

Challenge level

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