Geometric sequences

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
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
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
Clickety click and all the sixes

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
Sixty-seven squared

Age

16 to 18

Challenge level

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?
problem
Smaller and smaller

Age

7 to 14

Challenge level

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?
problem
Ruler

Age

16 to 18

Challenge level

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?
problem
Tiny nines

Age

14 to 16

Challenge level

What do you notice about these families of recurring decimals?
problem
Transformations tables

Age

7 to 11

Challenge level

These grids are filled according to some rules - can you complete them?
problem
The tomato and the bean

Age

5 to 7

Challenge level

At the beginning of May, Tom put his tomato plant outside. On the same day he sowed a bean in another pot. When will the two be the same height?