Converging Product

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)... ?

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?

Binary Squares

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

Summing Geometric Progressions

Why do this problem?

This problem provides an introduction to summing geometric series, and allows students to discover for themselves the formulae used to calculate such sums. By seeing a particular case, students can perceive the structure and see where the general method for summing such series comes from.

Possible approach

You may wish to show the video, in which Alison works out the sum of the first twenty terms of $2, 8, 32, 128, 512 ...$ in silence, or you may wish to recreate the video for yourself on the board.

"Can you make sense of the video? See if you can recreate it with your partner."
"Can you adapt the method to work out the sum of the first 50 terms of the sequence?"

Share answers and explanations of how they worked it out.

Next, give students the following questions:

• $3, 9, 27, 81, 243 ...$ up to the 15th term
• $5, 10, 20, 40, 80 ...$ up to the 12th term
• $\sum_{i=1}^{20}(3 \times 2^{i-1})$
• $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} ...$ up to the 10th term

"Can you adapt the method to work out these sums? In a while I'm going to give you another question like these and you'll need to be able to work it out efficiently"

While students are working, listen out for useful comments that they make about how to work out such sums generally. Then bring the class together to share answers and methods for the questions they have worked on.

Make up a few questions like those above, and invite students out to the board to work them out 'on the spot', explaining what they do as they go along.

Next, invite students to create a formula from their general thinking:

"Imagine a sequence with first term $a$ and each term after that is multiplied by $r$. Can you use what you did with the numerical examples to create a formula for the sum of the series?"

Give students time to think and discuss in pairs and then share their suggestions.

To finish off, perhaps offer some questions for students to try out their formula, or discuss what happens when the sequence has infinitely many terms, and the conditions necessary for convergence.

Key questions

How does Alison sum the series in the video? What are the key steps?
What do you need to do to adapt Alison's method for other sequences?

Possible extension

Telescoping Series invites students to investigate another family of series and how to sum them.

Possible support

There is a proof sorter activity that could be offered to students who are having difficulty generalising the method.