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#### Resources tagged with Limits of Sequences similar to Happy Numbers:

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

### 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.

### 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?

### 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?

### Summing Geometric Progressions

##### Stage: 4 and 5 Challenge Level:

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

### Approximating Pi

##### Stage: 4 and 5 Challenge Level:

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?

### Continued Fractions I

##### Stage: 4 and 5

An article introducing continued fractions with some simple puzzles for the reader.

### Archimedes and Numerical Roots

##### Stage: 4 Challenge Level:

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?