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?
What might this notation mean?
Can I write these as more conventional fractions? In other words, what do they equal?
The first term in the sequence is just \(1\).
The next is \(1 + \cfrac{1}{1} = 2\) in a familiar way.
Then we have a less familiar expression, but we can work through it ‘from the bottom up’ to make sense of it. We have \[1 + \cfrac{1}{1 + \cfrac{1}{1}} = 1 + \cfrac{1}{2} = \frac{3}{2}.\]
And then, similarly, \[1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1}}} = 1 + \cfrac{1}{1 + \cfrac{1}{2}} = 1 + \cfrac{2}{3} = \frac{5}{3}.\]
How do I expect the sequence to continue?
Does the sequence have a last term?
If I look at the original form of the sequence, I can imagine it going on forever, with more and more layers. I might need to think a bit more about what the conventional fractions look like as the sequence continues.
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?
It might be helpful to study a few more terms of the sequence, and then to collect the information in a way that helps us to make sense of it.
| Term of sequence | Fraction | Decimal |
|---|---|---|
| \(1\) | \(\dfrac{1}{1}\) | \(1\) |
| \(1 + \cfrac{1}{1}\) | \(\dfrac{2}{1}\) | \(2\) |
| \(1 + \cfrac{1}{1 + \cfrac{1}{1}}\) | \(\dfrac{3}{2}\) | \(1.5\) |
| \(1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1}}}\) | \(\dfrac{5}{3}\) | \(1.666\dots\) |
| \(1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1}}}}\) | \(\dfrac{8}{5}\) | \(1.6\) |
| \(1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1}}}}}\) | \(\dfrac{13}{8}\) | \(1.625\) |
| \(1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1}}}}}}\) | \(\dfrac{21}{13}\) | \(1.615\dots\) |
| \(1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1}}}}}}}\) | \(\dfrac{34}{21}\) | \(1.619\dots\) |
It looks as though the terms alternately get larger and smaller, and it looks as though they might be tending to a limit that’s \(1.61...\) (not sure what the other decimal places might be yet).
We could try to think carefully about whether the terms will continue to alternately get larger and smaller. You could explore this further in Comparing continued fractions.
I’d like to know whether the sequence really does tend to a limit. If it does, then it ought to be represented by \[1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \dots}}}}}.\] If I call that \(x\), then I could notice that \(x\) seems to reappear inside itself: \[x = 1 + \frac{1}{x}.\] And now I can rearrange to get a quadratic: it’s the same as the equation \[x^2 - x - 1 = 0.\] I can solve that using the quadratic formula. I find that \[x = \frac{1 \pm \sqrt{5}}{2},\] and since clearly \(x\) is positive I know that \[x = \frac{1 + \sqrt{5}}{2}.\] So it seems that the sequence really does tend to a limit, and that limit is \(\frac{1 + \sqrt{5}}{2} = 1.618...\) (which fits with my observations from the table above). In fact, it looks as though the sequence is alternately larger and smaller than this limit.
Can I efficiently work out the value of each term of the sequence?
I got bored while working out the table, so I found myself looking for shortcuts. I noticed that rather than working from the bottom up each time, I could use the previous term in the sequence: roughly speaking, \[\textrm{new term } = 1 + \frac{1}{\textrm{last term}}.\]
I can be a bit more formal if I use some symbols. If I know that one term of the sequence is \(\frac{a}{b}\), then the next is \[1 + \frac{1}{\bigl(\frac{a}{b}\bigr)} = 1 + \frac{b}{a} = \frac{a + b}{a}.\]
I wrote brackets around the \(\frac{a}{b}\) in the first fraction, so that I know what the numerator and denominator are. If I’d written \(\dfrac{1}{\frac{a}{b}}\) without brackets, I might have got confused and thought it meant \(\dfrac{\bigl(\frac{1}{a}\bigr)}{b}\).
So to find the numerator of the next term, I add the numerator and denominator of the last term, and to find the denominator of the next term, I simply take the numerator of the last term. And I can check that fits with the terms I found in my table.
Do I notice anything interesting about the numbers in the fractions?
I happen to recognise that each term in the sequence is the ratio of consecutive Fibonacci numbers. I can see why that is true using the reasoning above, that tells me how to find one term from the next.
So I’ve learned that the ratio of one consecutive Fibonacci number to the previous one converges to the golden ratio, \(\frac{1 + \sqrt{5}}{2}\).
The second sequence was: \[1, \quad 1 + \cfrac{1}{2}, \quad 1 + \cfrac{1}{2 + \cfrac{1}{2}}, \quad 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2}}}, \quad \dotsc .\]
Using similar ideas, can you find what the sequence converges to?
