The terms of a sequence are given by this equation where each term is found from the two preceding terms.

.For example if the first two terms are 2 and 5 then the next term is (1+5)/2=3 and the sequence is

2,  5,  3,  4/5,  3/5,  2,  5, … etc.

Investigate the sequences you get when you choose your own first two terms. What happens to your sequences?

Make a conjecture about the behaviour of sequences given by this rule.

Can you prove your conjecture?

See the Notes for Teachers.

This problem is adapted from the NRICH task Loopy with permission of the University of Cambridge. All rights reserved.

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