• Support PF! Buy your school textbooks, materials and every day products Here!

Introductory Analysis: Inductively define a sequence Sn

  • Thread starter fatfatfat
  • Start date
  • #1
16
0

Homework Statement




Let S1=1 and inductively define the sequence Sn so that Sn+1 = [tex]\sqrt{Sn + 1}[/tex]


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 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?
 

Answers and Replies

  • #2
761
13
Surely this isn't the complete assignment . Please post the entire question only then I can help you.
 
  • #3
16
0
Let S1=1 and inductively define the sequence (Sn) so that Sn+1 = [tex]\sqrt{Sn + 1}[/tex] for n[tex]\in[/tex] 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)=[tex]\frac{1}{2}[/tex] (1 + [tex]\sqrt{5}[/tex] )
 
  • #4
16
0
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
Dick
Science Advisor
Homework Helper
26,258
618
'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
16
0
Oh, haha. Thank you.
 

Related Threads for: Introductory Analysis: Inductively define a sequence Sn

Replies
4
Views
5K
Replies
2
Views
1K
  • Last Post
Replies
5
Views
998
  • Last Post
Replies
16
Views
404
  • Last Post
Replies
8
Views
1K
Replies
4
Views
939
Replies
2
Views
2K
  • Last Post
Replies
2
Views
849
Top