Ask Me Help Desk - Recurrence relations

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Recurrence relations

Hello can someone please help with this question?

Consider the sequence an+1 =sqrt(2 + an)

Show that for all n in natural numbers.

n+1 - a= (a_n=-a_n-1)/(sqrt(2+a_n+sqrt(2+a_n-1))

I really don't understand what to do?

I think what you mean is this:

$a_{n+1} - a_n = \frac {a_n - a_{n-1}}{\sqrt{2+a_n}+\sqrt{2+a_{n-1}}}$

It's simply a matter of using the definition, and incrementing backwards from $a_{n+1}$ to $a_n$ . To get you started, from the definition of $a_{n+1}$ you get:

$a_n = \sqrt{2 + a_{n-1}}$

so:

$a_{n+1}-a_n = \sqrt{2 + a_n} - \sqrt{2 + a_{n-1}}$

Can you take it from here?

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