### Telescoping Series

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

### Degree Ceremony

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

### OK! Now Prove It

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

# Vanishing Point

### Why do this problem

When meeting geometric series for the first time, there's a temptation to look at the algebraic definitions first and derive the summation formula without giving students the opportunity to get to grips with why some sums converge and some diverge. In this problem, by offering a geometrical representation, we hope it will be clear to students why these series converge, while providing a hook to inspire curiosity into exploring other convergent series.

### Possible approach

You may wish to spend some time with students working on Diminishing Returns before introducing this problem.

Show the interactivity from the problem, and gradually move the slider back and forth a few times to show the pattern growing.

"What proportion of the image is coloured in each colour at each stage?"
"How might the pattern continue?"
Give students some time to discuss and work out their answers, and then bring the class together to agree on a sum that represents the shades areas.

"What would happen if the pattern carried on forever?"
Again, give students some thinking time, and then share different ways of perceiving that $\frac13$ of the final pattern would be shaded blue and $\frac23$ purple. There are some suggestions in the problem of how students might convince themselves, but there are also many other ways of visualising the sum.

Next, show the other patterns and invite students to carry out the same process of working out the sum of fractions represented by the blue shaded areas at each stage, and the total area shaded blue in the completed pattern. Again, encourage a variety of different ways of visualising the sum.

Finally, after discussing the sums and limits for the patterns given in the problem, you might like to invite students to create their own images to represent different sums of fractions.

### Key questions

What fraction of the square is blue at each stage?
How can you visualise the pattern to help you to find the limit?

### Possible extension

The patterns could be used as a brief introduction to the idea of self-similarity in fractal patterns.

### Possible support

There are suggestions in the Teachers' Resources to Diminishing Returns which might be useful to support students who are struggling with this task.