Proving the Convergence of a Sequence Defined by Induction

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Discussion Overview

The discussion revolves around proving the convergence of a sequence defined by induction, specifically a sequence defined as z0=3 and zk=(zk-1)² for integers k ≥ 1. Participants are attempting to show that the sequence can be represented as Ci=32i for i ≥ 0, while clarifying the definitions and relationships between the variables involved.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses uncertainty about the meaning of Ci and whether they are correct in trying to show that Zk = Ci.
  • Another participant suggests using "Strong Induction" instead of basic mathematical induction.
  • There is confusion about the introduction of Ci, with participants questioning its definition and relevance to the problem.
  • A participant attempts to prove that Ck+1=32k+1 using the induction hypothesis but is unsure if their reasoning is correct.
  • Clarification is sought regarding the definition of Ci, with one participant suggesting it might just be another representation of Zk.
  • A later reply indicates that a professor clarified the notation should be Zi instead of Ci, which may simplify the problem.
  • One participant expresses a desire to continue working on the problem, considering changing Cs to Zs for clarity.

Areas of Agreement / Disagreement

Participants generally agree that there is confusion regarding the notation and definitions used in the problem. However, there is no consensus on the correct approach or resolution of the problem itself.

Contextual Notes

The discussion highlights limitations in understanding the definitions of variables and the application of induction methods. There are unresolved questions about the correct notation and its implications for the proof.

johnnyICON
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A sequence z0,z1,z2,... is defined by letting z0=3, and zk=(zk-1)2 for all integers k greater than equal to 1. Show that Ci=32i for i greater than or equal to 0.

I wasn't to clear on what it meant by this, so what I have is that I am trying to show that Zk = Ci. Is that correct?

From there, I proved the base case to be true.

Proving n+1 to be true is where I am having problems.
Cn+1=(Cn)2 and
Zn+1=32n+1=32n(2)

I don't see how I can express Cn+1 to be like Zn+1. I'm not even sure if I am understanding the problem correctly. Am I at least going in the right direction? Any hints?
 
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Is this to be done using "Strong Induction"
I was using basic mathematical induction.
 
johnnyICON said:
I wasn't to clear on what it meant by this, so what I have is that I am trying to show that Zk = Ci. Is that correct?


We dont' know. You're the one that introduced Ci without explaining what it is.
 
Here's how far I've gotten now,
I'm trying to show that Ck+1=32k+1.

By definition,
Ck+1
= (Ck)2
= (32k)2 By the Induction Hypothesis
= (32k(2))
= (32k+1)

Is that correct?
 
matt grime said:
We dont' know. You're the one that introduced Ci without explaining what it is.

The very first sentence is straight from my textbook. I'm guessing Ci is just another way of representing Zk.
 
johnnyICON said:
The very first sentence is straight from my textbook. I'm guessing Ci is just another way of representing Zk.
"The very first sentence" was "A sequence z0,z1,z2,... is defined by letting z0=3, and zk=(zk-1)2 for all integers k greater than equal to 1." which says nothing about Ci. You can't prove anything about Ci without knowing exactly how it is defined!
 
"A sequence z0,z1,z2,... is defined by letting z0=3, and zk=(zk-1)2 for all integers k greater than equal to 1. Show that Ci=32i for i greater than or equal to 0."

I e-mailed my professor, he said it is supposed to be Zi, not C... I don't know if that helps...
 
It helps and makes it easy; just do it. And if you can't write saying where you're stuck.
 
johnnyICON said:
Here's how far I've gotten now,
I'm trying to show that Ck+1=32k+1.

By definition,
Ck+1
= (Ck)2
= (32k)2 By the Induction Hypothesis
= (32k(2))
= (32k+1)

Is that correct?

I'm still fixated on this. :biggrin: Maybe if I make the Cs Zs instead?
 

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