Proving the Convergence of a Sequence Defined by Induction

[johnnyICON]

A sequence z₀,z₁,z₂,... is defined by letting z₀=3, and z_k=(z_k-1)² for all integers k greater than equal to 1. Show that C_i=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 Z_k = C_i. 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.
C_n+1=(C_n)² and
Z_n+1=3²ⁿ⁺¹=3²ⁿ⁽²⁾

I don't see how I can express C_n+1 to be like Z_n+1. I'm not even sure if I am understanding the problem correctly. Am I at least going in the right direction? Any hints?

 

[johnnyICON]

Is this to be done using "Strong Induction"
I was using basic mathematical induction.

 

[matt grime]

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.

 

[johnnyICON]

Here's how far I've gotten now,
I'm trying to show that C_k+1=3^2k+1.

By definition,
C_k+1
= (C_k)²
= (3^2k)² By the Induction Hypothesis
= (3^2k(2))
= (3^2k+1)

Is that correct?

 

[johnnyICON]

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.

 

[HallsofIvy]

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 C_i. You can't prove anything about C_i without knowing exactly how it is defined!

 

[johnnyICON]

"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 Z_i, not C... I don't know if that helps...

 

[matt grime]

It helps and makes it easy; just do it. And if you can't write saying where you're stuck.

 

[johnnyICON]

Here's how far I've gotten now,
I'm trying to show that C_k+1=3^2k+1.

By definition,
C_k+1
= (C_k)²
= (3^2k)² By the Induction Hypothesis
= (3^2k(2))
= (3^2k+1)

Is that correct?

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