Question

To what value does \[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}}\]converges?

Answer 1

The back of the envelope way is to first assume it does converge to some value, call it x. In which case
\[ x = \sqrt{1 + x} \]
Then solve for x.

But you also need to convince yourself that it does indeed converge so that this isn't' a nonsense calculation.

Answer 2

seems to be 1.6180339887499

Answer 3

Ugh. Closed form answer please!

Answer 4

What's a closed form answer?

Answer 5

oh you mean like an expression with a big sigma thing in it?

Answer 6

you've written down an approximation of the answer. Completely unnecessary. You can write an exact, closed form of the solution.

Answer 7

x=1/2(1+sqrt5) is the solution of x=sqrt(1+x).

Answer 8

But how does this help?

Answer 9

\[x = \frac{\sqrt{5}+1}{2}\]

Answer 10

isn't that the golden ratio?

Answer 11

why is it the golden ratio?

Answer 12

There's a puzzle for you.

Answer 13

:(

Answer 14

I don't even know what a golden ratio is.

Answer 15

back of the envelop! i like it. actually hard part is showing it converges, easy part is solving a quadratic equation.

Answer 16

you can google 'golden ratio' and get more information that you can possibly use.

Answer 17

def fib(n):
last = 0
this = 1
if n < 2:
return n
for i in xrange(1,n):
new = last + this
last = this
this = new
return this
for i in xrange(1,101):
print float(fib(i+1))/float(fib(i))

it also prints out the golden ratio

Answer 18

i can't really read this but it looks like what you have is the limit of the quotients of the fibonnaci numbers, which is the golden ratio

Answer 19

Hmm.... this all seems deja vu to me

Answer 20

Excercise try to find X in here $$ X=5^0+ \sqrt{5^1+\sqrt{5^2+\sqrt{5^4+\sqrt{5^8+\sqrt{5^{16}+\sqrt{5^{32}+\dots}}}}}}$$

Answer 21

That's a nice competition question.

Answer 22

I think the solution can be implemented using python

Answer 23

this should be in project euler

Answer 24

It can be solved--and should be--with pencil and paper.

Answer 25

alright. what's step 1?

Answer 26

Think about it, given your original question. They're very related.

Answer 27

alright what's the trick?

Answer 28

It is easy when you know the trick.

Answer 29

the trick is that the next number is equal to the previous number?

Answer 30

or something?

Answer 31

think about it for a while.

Answer 32

is it the same answer but multiplied by 5?

Answer 33

if it is, explain why. If not, what is it. Do some work.

Answer 34

btw @JamesJ technically, you need to prove the convergence before writing $$ x = \sqrt{1 + x} $$ isn't ? ;)

Answer 35

lol okay. I'll fire up google and if it doesn't help, I will think about it.

Answer 36

@FFM. That is exactly what I said.

Answer 37

By Google-ing you may be robing yourself of an opportunity to solve a good problem :)

Answer 38

OH sorry I haven't read all of the discussions.

Answer 39

is the answer 5?

Answer 40

nice trick!

Answer 41

$$ \frac{7+\sqrt{5}}{2}$$

Answer 42

Yes, that's right.

Answer 43

nooo you revealed the answer to me :(

Answer 44

Now figure out why it's right.

Answer 45

\[X = 1 + \sqrt{5 + X}\]

Answer 46

almost there

Answer 47

I computed the first few terms and found that (call the first summation Y):
\[X=1+\sqrt{5}Y.\]

Answer 48

it's been half an hour, and no luck for me :(

Answer 49

\[\sqrt{5+\sqrt{5^2}}=\sqrt{5+5}=\sqrt{5}\sqrt{2}.\]
\[\sqrt{5+\sqrt{5^2+\sqrt{5^4}}}=\sqrt{5+\sqrt{5^2+5^2}}=\sqrt{5+5\sqrt{2}}=\sqrt{5}\sqrt{1+\sqrt{2}}.\]
\[\sqrt{5+\sqrt{5^2+\sqrt{5^4+\sqrt{5^8}}}}=\sqrt{5+\sqrt{5^2+\sqrt{5^4+5^4}}}\]
\[=\sqrt{5+\sqrt{5^2+5^2\sqrt{2}}}=\sqrt{5+5\sqrt{1+\sqrt{2}}}=\sqrt{5}{\sqrt{1+\sqrt{1+\sqrt{2}}}}\].
Does this help?

Answer 50

kind-of.... I still can't find that recurrence relation thing I'm supposed to look for

Answer 51

I think you can see that\[X=5^0+\sqrt{5}Y.\]

Answer 52

yes

Answer 53

but I still can't solve it

Answer 54

\[X=1+\sqrt{5}.{\sqrt{5}+1 \over 2}=1+{5+\sqrt{5} \over 2}={7+\sqrt{5} \over 2}.\]

Answer 55

I wouldn't be able to find it if James didn't do the first part though.

Answer 56

*Agdgdgdgwngo feels enlightened.

Answer 57

But how did you figure that trick out?

Answer 58

*Agdgdgdgwngo is actually enlightened.

Answer 59

so this is how we apply the golden ratio