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Mathematical Induction using a strong hypothesis

1. The problem statement, all variables and given/known data
If a0=1, and a1=2, and

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



2. Relevant equations



3. The attempt at a solution
(B) a0=1=2^0=1 yes is true
a1=2=2^1=2 yes is true

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

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)=
(2k)^2(k-1)=2(k+1)
Please someone help if I am doing something wrong. I will appreciate. Thank you.
 
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It seems that you have the right idea, but I do not know what is different between (I) and your solution for ak+1. Where precisely is your inductive hypothesis?
 

Dick

Science Advisor
Homework Helper
26,255
618
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.
 
That is where I am confused. I know that I need to fallow the form an=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:smile:
 
Thank you guys.
 

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