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This year we learned this method of characteristic equation to find the formula for the general element of a string of numbers (i.e. $a_n$)

the characteriatic equation of the recurrence relation $a_n=p_1a_{n-1}+...+p_ka_{n-k}$ is: $t^k=p_1t^{k-1}+...+p_kt^0$. My question is very vague: how is this method correct? I'm not saying I can't find the truth in it, I just can't understand how one can reach the descovery of this method. More accurately, what is the analogy behind it? Why is it used a polynomial in help for its solvation? I hope I made my point one way or another and that you could help me to find out the mistery behind it. Thanks in advance!

Anonymus
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    The key remark is that, if $\lambda$ is a root of the characteristic polynomial, then $a_n=\lambda^n$ satisfies the recursion. Just write it down! (Note: if the characteristic equation has multiple roots then the technique gets a little less clear). – lulu Oct 02 '17 at 19:07

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