Consider the sequence \[1, \quad 1 + \cfrac{1}{1}, \quad 1 + \cfrac{1}{1 + \cfrac{1}{1}}, \quad 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1}}}, \quad \dotsc .\]
What do you make of it?
In playing around with the sequence you may well have asked some questions.
Here are some questions we asked about these sequences that you might like to think about, if you have not already considered them.
What might this notation mean?
Can I write these as more conventional fractions? In other words, what do they equal?
How do I expect the sequence to continue?
Does the sequence have a last term?
Can I predict what happens to the terms of the sequence a long way down the line?
Are the terms in the sequence getting larger or smaller?
Can I efficiently work out the value of each term of the sequence?
Do I notice anything interesting about the numbers in the fractions?
