Proving the Convergence of a Sequence Defined by Induction

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In summary: So, I'm trying to show that Zk+1=32k+1.In summary, the sequence Zk is defined by letting z0=3, and zk=(zk-1)2 for all integers k greater than equal to 1. To prove that Zi=32i for i greater than or equal to 0, we use induction and show that Zk+1=32k+1. By definition, Zk+1=(Zk)2. By the induction hypothesis, (Zk)2=(32k)2. This can be rewritten as (32k(2)), which is equal to (32k+1). Therefore, we have shown that Zk+1=32
  • #1
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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  • #2
Is this to be done using "Strong Induction"
I was using basic mathematical induction.
 
  • #3
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.
 
  • #4
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?
 
  • #5
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.
 
  • #6
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!
 
  • #7
"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...
 
  • #8
It helps and makes it easy; just do it. And if you can't write saying where you're stuck.
 
  • #9
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?
 

1. What is induction and why is it important in science?

Induction is a method of reasoning that uses specific observations to make generalizations or predictions about a larger set of phenomena. It is important in science because it allows us to make hypotheses and theories based on evidence and observations.

2. What are some common challenges or issues with using induction in scientific research?

One potential challenge with induction is that it relies on a limited number of observations, which may not accurately represent the entire population. Additionally, there is always a possibility for new evidence to contradict or disprove the initial generalization made through induction.

3. How can we minimize the potential errors or biases in using induction in scientific studies?

One way to minimize errors in induction is to use a larger and more diverse sample size for observations. This can help to reduce the impact of outliers and increase the reliability of the generalization. It is also important to constantly evaluate and reevaluate the evidence and conclusions made through induction.

4. Can induction be used in all areas of science?

Yes, induction can be used in all areas of science as it is a fundamental method of reasoning. However, the degree to which it is used may vary depending on the type of research and the availability of evidence.

5. How does deduction differ from induction in scientific research?

Deduction is a method of reasoning that starts with general principles or theories and uses these to make specific predictions or observations. In contrast, induction starts with specific observations and uses them to make generalizations or predictions. Deduction is often used to test or validate hypotheses made through induction.

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