### There are 20 results

Broad Topics >

Patterns, Sequences and Structure > Geometric sequences

##### 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.

##### Age 14 to 16 Challenge Level:

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.

##### 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)... ?

##### Age 14 to 18 Challenge Level:

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

##### Age 16 to 18 Challenge Level:

If a number N is expressed in binary by using only 'ones,' what can
you say about its square (in binary)?

##### Age 14 to 18 Challenge Level:

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

##### Age 16 to 18 Short Challenge Level:

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

##### Age 16 to 18

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.

##### 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?

##### 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?

##### 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?

##### 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.

##### 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!

##### 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?

##### Age 14 to 16 Challenge Level:

Simple additions can lead to intriguing results...

##### 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?

##### 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.

##### Age 16 to 18 Challenge Level:

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.

##### Age 14 to 16 Challenge Level:

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

##### Age 16 to 18 Challenge Level:

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