### Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

### Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

### Just Opposite

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

# Double Trouble

##### Age 14 to 16Challenge Level

Charlie has been adding fractions in the sequence $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots$ where each fraction is half the previous one:

$$\frac{1}{2} + \frac{1}{4}$$ $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8}$$ $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} +\frac{1}{16}$$

Work out the answers to Charlie's sums. What do you notice?

Will the pattern continue?
How do you know?

Try writing an expression for $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^n}$$

Could you convince someone else that your expression is correct for all values of $n$?

Charlie drew a diagram to try to explain what was going on:

Use Charlie's diagram to explain why $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^n} = 1-\frac{1}{2^n} = \frac{2^n-1}{2^n}$$

Alison has been adding numbers in the sequence $1, 2, 4, 8, \dots$ where each number is twice the previous one:

$$1 + 2$$ $$1 + 2 + 4$$ $$1 + 2 + 4 + 8$$

Work out the answers to Alison's sums. What do you notice?

Will the pattern continue?

How do you know?

Try writing an expression for $$1 + 2 + 4 + \dots + 2^n$$

Could you convince someone else that your expression is correct for all values of $n$?

Alison drew a diagram to try to explain what was going on:

Can you use Alison's diagram to explain why $$1 + 2 + 4 + \dots + 2^n = 2^{n+1}-1$$