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# Find the golden ratio in surd form

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1. I have this question:
The golden ratio appears in many natural situations. It can be found by looking at the ratio of success in terms of the Fibonacci sequence:
1,1,2,3,5,8,13,21,34
2/1, 3/2, 5/3, 8/5
2, 1.5, 1.67, 1.6... the golden ratio is about equal to 1.6.
Find it in surd form.

So far I've figured that 2 should be a, then 3 should be b, 5 should be a+b, and so on. But I need help on how I can make this into a quadratic maybe and express it as a surd. Thanks...
2. (Original post by Yewdraconis)
I have this question:
The golden ratio appears in many natural situations. It can be found by looking at the ratio of success in terms of the Fibonacci sequence:
1,1,2,3,5,8,13,21,34
2/1, 3/2, 5/3, 8/5
2, 1.5, 1.67, 1.6... the golden ratio is about equal to 1.6.
Find it in surd form.

So far I've figured that 2 should be a, then 3 should be b, 5 should be a+b, and so on. But I need help on how I can make this into a quadratic maybe and express it as a surd. Thanks...
I don't think you will get anywhere with that.

I've seen something like this at Cambrige over a year ago, but they started later down with my method, so I think my method is correct however I don't have much experience with these recurrence things and infinities, so I'll give a shot at explaining this. Anyway:

Second sequence. Observe:

So:

We can observe that term, with , is given by:

for

Now we get the golden ratio as we go further and further down the sequence, so

Meaning

As this sequence converges onto a particular value (which is our golden ratio!), it is safe to assume that after an infinite amount of steps, we get:

...as the difference between and becomes 0 at infinity, hence they are the same, referring to this value as .

At this point just solve the equation and you get (take the positive value, as our sequence is strictly positive).

Feel free to ask if you don't understand something.
Anyone can also feel free to correct me if I made a mistake, not too familiar with recurrence relations and limits combined into one as I said.
3. (Original post by Yewdraconis)
I have this question:
The golden ratio appears in many natural situations. It can be found by looking at the ratio of success in terms of the Fibonacci sequence:
1,1,2,3,5,8,13,21,34
2/1, 3/2, 5/3, 8/5
2, 1.5, 1.67, 1.6... the golden ratio is about equal to 1.6.
Find it in surd form.

So far I've figured that 2 should be a, then 3 should be b, 5 should be a+b, and so on. But I need help on how I can make this into a quadratic maybe and express it as a surd. Thanks...

let the nth term be b and the n+1th term be a

n+2th term=a+b

the limit of the sequence is when the ratio of the terms n to n+1 and n+1 to n+2 is the same

a/b=(a+b)/a
a/b=1+b/a
let a/b=x
x=1+1/x
solve for x and the solutions will be the forward ratio and the backward ratio
RDKgames's way is a lot better and works for far more types of sequences but I think this one is a bit easier to read
4. (Original post by RDKGames)
I don't think you will get anywhere with that.

I've seen something like this at Cambrige over a year ago, but they started later down with my method, so I think my method is correct however I don't have much experience with these recurrence things and infinities, so I'll give a shot at explaining this. Anyway:

Second sequence. Observe:

So:

We can observe that term, with , is given by:

for

Now we get the golden ratio as we go further and further down the sequence, so

Meaning

As this sequence converges onto a particular value (which is our golden ratio!), it is safe to assume that after an infinite amount of steps, we get:

...as the difference between and becomes 0 at infinity, hence they are the same, referring to this value as .

At this point just solve the equation and you get (take the positive value, as our sequence is strictly positive).

Feel free to ask if you don't understand something.
Anyone can also feel free to correct me if I made a mistake, not too familiar with recurrence relations and limits combined into one as I said.
Thanks this makes some sense, I think I can work the rest out. Thanks

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