### Triangle Incircle Iteration

Keep constructing triangles in the incircle of the previous triangle. What happens?

### Loopy

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

### Converging Means

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the two sequences.

# Peaches in General

##### Age 14 to 16 Challenge Level:

Obviously start by doing the original problem, that's plenty hard enough as a first challenge if you haven't seen anything like it before, but after that ask yourself the question:

• What things might generalise?

Perhaps the fraction taken each day, or the number of days , or the 'plus one more', or something else, but take those possibilities one at a time.

If you use a spreadsheet (and there are really strong reasons for doing that in this type of problem), what do you need to calculate in each column of the sheet? You can have as many columns as you want, so do simple calculations, then calculations further along the row that use those answers, rather than complicated calculations in single cells.

Will you make your first column a number of peaches before any eating happens? It might be better to have the number of peaches in the final column, making your row a calculation trail that deduces the number at the start?