Mathematical Induction using a strong hypothesis

[mamma_mia66]

Homework Statement

If a₀=1, and a₁=2, and

a_n=(a_(n-1))^2/a_n-2 for n>=2,
prove by induction that a_n=2^n for n>=0

Homework Equations

The Attempt at a Solution

(B) a₀=1=2^0=1 yes is true
a₁=2=2^1=2 yes is true

(I) a_k=(2^k-1)^2/2^k-2=2^k

Is it true that what I solved. It seems very easy.

I started first with a_(k+1)=(a_(k+1-1))^2/a_(k+1-2)=
(2^k)^2^(k-1)=2^(k+1)
Please someone help if I am doing something wrong. I will appreciate. Thank you.

 

[Tedjn]

It seems that you have the right idea, but I do not know what is different between (I) and your solution for a_k+1. Where precisely is your inductive hypothesis?

 

[Dick]

It IS very easy. So your induction step is: if a_{k-1}=2^(k-1) and a_{k-2}=2^(k-2) then a_k=2^(k) (by doing the algebra you doubtless did). It looks fine to me.

 

[mamma_mia66]

That is where I am confused. I know that I need to fallow the form a_n=2^n

Then I don't need to do a_(k+1)...

I did so many problems in my HW and now I don't get it what I am doing

 

[mamma_mia66]

Thank you guys.