# Homework Help: Introductory Analysis: Inductively define a sequence Sn

1. Jan 25, 2009

### fatfatfat

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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 Sn by using Sn+1.

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

How should I go about starting this?

2. Jan 25, 2009

### dirk_mec1

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

3. Jan 25, 2009

### fatfatfat

Let S1=1 and inductively define the sequence (Sn) so that Sn+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 + $$\sqrt{5}$$ )

4. Jan 25, 2009

### 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.

5. Jan 25, 2009

### 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.

6. Jan 25, 2009

### fatfatfat

Oh, haha. Thank you.

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