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February 21st, 2019, 10:46 PM  #1 
Member Joined: Nov 2012 Posts: 80 Thanks: 1  Recurrence relation
For second order linear homogeneous recurrence relation with constant coefficients, is it that the general sequence is only in the form of 1, t, t^2, t^3, ..., t^n or just one of the general sequences which satisfies the characteristic equation? Thank you.

February 21st, 2019, 11:35 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,379 Thanks: 2011 
See this article.

February 22nd, 2019, 12:13 AM  #3 
Member Joined: Nov 2012 Posts: 80 Thanks: 1 
Thank you for the reference. It seems that the characteristic equation for a second order linear homogeneous recurrence relation with constant coefficients should be a Quadratic equation.

February 22nd, 2019, 06:40 AM  #4 
Member Joined: Nov 2012 Posts: 80 Thanks: 1 
Can I say that it is first assumed that the indexed term a(n)=t^n, and it happens that the assumption satisfies the recurrence relation by solving the characteristic equation?

February 22nd, 2019, 06:49 AM  #5 
Senior Member Joined: Dec 2015 From: iPhone Posts: 440 Thanks: 68 
Yes, it turns into a polynomial equation. But I donâ€™t know the reason behind it; let someone else prove it or give an article of proof. Last edited by skipjack; February 22nd, 2019 at 09:31 AM. 
February 22nd, 2019, 09:29 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,379 Thanks: 2011 
If the characteristic equation has distinct roots p and q, the nth term of the sequence is a(p^n) + b(q^n), where a and b are constants. 

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