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Broad Topics > Sequences, Functions and Graphs > Limits of Sequences

### Zooming in on the Squares

##### Stage: 2 and 3

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

### Litov's Mean Value Theorem

##### Stage: 3 Challenge Level:

Start with two numbers. This is the start of a sequence. The next number is the average of the last two numbers. Continue the sequence. What will happen if you carry on for ever?

### Happy Numbers

##### Stage: 3 Challenge Level:

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

### Light Blue - Dark Blue

##### Stage: 2 Challenge Level:

Investigate the successive areas of light blue in these diagrams.