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Mathematical induction :

Posted June 12th, 2009 by alpha
in

Given that $ u_{n+1}=3u_n -2u_{n-1} $ and $ u_0=2 ,u_1=3 $ .Prove that $ u_n=2^n+1 $ for all $ n\in N $

Steffensen's inequality proof ›

Comments

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On January 5th, 2012 MSA says:

Proof without induction:
$ u_{n+1}-3u_{n}+2u_{n-1}=0 $

=> $ t^2-3t+2=0=>t_{1}=1,t_{2}=2 $

$ u_{n}=t_{1}^na+t_{2}^nb $
for n=0 and n=1 =>
a+ b=2
2a+b=3
=>a=b=1=>$ u_{n}=t_{1}^n+t_{2}^n=1+2^n $

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