Geometric sequences

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.

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?

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?

What do you notice about these families of recurring decimals?

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

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?