How can the recurrence formula for a sequence be found?

[Mathematicsss]

Homework Statement

Find a recurrence formula for the sequence (ai) = 1, sqrt3, sqrt(1+sqrt3), sqrt(1+sqrt(1+sqrt2)) in terms of i and ai

Homework Equations

The Attempt at a Solution

no idea where to start, this is a bonus question, and I have learned how to solve these type of problems

 

[jedishrfu]

I'd investigate nested radicals as this was a favorite of Ramanujan.

As a hint,$x = \sqrt{1 + \sqrt{1+ \sqrt{1 + ...}}}$ is basically the same as $x = \sqrt{1 + x}$

 

[Mathematicsss]

I'd investigate nested radicals as this was a favorite of Ramanujan.

As a hint,$x = \sqrt{1 + \sqrt{1+ \sqrt{1 + ...}}}$ is basically the same as $x = \sqrt{1 + x}$

That hasn't helped. Please explain.

 

[Mark44]

(ai) = 1, sqrt3, sqrt(1+sqrt3), sqrt(1+sqrt(1+sqrt2))

Shouldn't the last one you listed be $\sqrt{1 + \sqrt{1 + \sqrt 3}}$?
Start by listing the elements of your sequence in an organized fashion, like so:
$a_0 = 1$
$a_1 = \sqrt 3$
$a_2 = \sqrt{1 + \sqrt 3}$ What is $a_2$ in terms of $a_1$?
$a_3 = \sqrt{1 + \sqrt{1 + \sqrt 3}}$ What is $a_3$ in terms of $a_2$?
Can you predict what $a_4$ is? If you can, you might be able to write $a_n$ in terms of $a_{n - 1}$, which is what you need to do for this problem.