Introductory Analysis: Inductively define a sequence Sn

[fatfatfat]

Homework Statement

Let S₁=1 and inductively define the sequence S_n so that S_n+1 = \sqrt{Sn + 1}

Homework Equations

The Attempt at a Solution

I'm not sure what it means to "inductively define".

I think it wants me to come up with an equation for S_n by using S_n+1.

Does it want me to define S_n in terms of S_n+1 or just in terms of n?

How should I go about starting this?

 

[dirk_mec1]

Surely this isn't the complete assignment . Please post the entire question only then I can help you.

 

[fatfatfat]

Let S₁=1 and inductively define the sequence (S_n) so that S_n+1 = \sqrt{Sn + 1} for n\in Natural Numbers.

(a) Prove that Sn is a monotonically increasing sequence.
(b) Prove that Sn is a bounded sequence.
(c) Prove that Sn converges.
(d) Prove that lim(Sn)=\frac{1}{2} (1 + \sqrt{5} )

 

[fatfatfat]

I'm sorry. That's the whole thing now.

I didn't realize that you needed the a,b,c,d parts to do the first part, I thought you had to inductively define Sn and then, using that definition, do the rest.

 

[Dick]

'Inductively define' doesn't mean you have to do anything. It's just pointing out that S_{n+1}=sqrt(S_n+1) is already an 'inductive' definition.

 

[fatfatfat]

Oh, haha. Thank you.