Introductory Analysis: Inductively define a sequence Sn

In summary, the question asks to define a sequence (Sn) with the initial term S1=1 and the recursive formula Sn+1 = \sqrt{Sn + 1}, and then prove that the sequence is monotonically increasing, bounded, and convergent with a limit of \frac{1}{2} (1 + \sqrt{5} ). The term "inductively define" simply means to use the recursive formula to define the sequence.
  • #1
fatfatfat
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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?
 
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  • #2
Surely this isn't the complete assignment . Please post the entire question only then I can help you.
 
  • #3
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
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
'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
Oh, haha. Thank you.
 

Related to Introductory Analysis: Inductively define a sequence Sn

1. What is a sequence?

A sequence is a list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term. The order of the terms is important and can be defined by an index, usually denoted as n.

2. How do you define a sequence inductively?

To define a sequence inductively, we start with the first term or initial value, and then we use a recursive formula to generate the subsequent terms. This means that each term is defined in terms of the previous term(s) in the sequence.

3. What is the notation for a sequence?

The notation for a sequence is usually written as (an) or {an}, where n represents the index or position of the term in the sequence. For example, (1, 3, 5, 7) or {2n}.

4. What is the difference between an arithmetic and geometric sequence?

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. For example, (2, 5, 8, 11) is an arithmetic sequence with a common difference of 3. A geometric sequence is a sequence where the ratio between consecutive terms is constant. For example, (2, 6, 18, 54) is a geometric sequence with a common ratio of 3.

5. How is a sequence represented graphically?

A sequence can be represented graphically by plotting the terms on a coordinate plane. The x-axis would represent the index n and the y-axis would represent the value of the term. Connecting the plotted points would form a line or curve, depending on the pattern of the sequence.

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