Choose any two numbers. The next number is the average of these two and the next is the average of the last two etc. Students can explore the limit - which is reached fairly quickly - either with a calculator or computer. The limit is a linear combination of the first two terms.
This has been called 'Littov's chain' but someone (Raz Lamplugh) told me this was a made up name! It generalises to e.g. the average of the last three numbers.