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I'm trying to solve a recurrence that looks like:

$$c_n x_n = x_{n-1} + \cdots + x_1$$

This looks simple, but the $c_n$ coefficient seems to make typical approaches to linear recurrences inapplicable. Is there a general approach that can be used for recurrences of this form?

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    We would need to know a little bit more about $c_n$. – 2017-02-10

1 Answers 1

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Hint:  subtracting two consecutive relations gives:

$$ \require{cancel} c_n x_n - c_{n-1}x_{n-1}= (x_{n-1} + \cancel{x_{n-2}+\cdots + x_1})-(\cancel{x_{n-2}+\cdots + x_1})=x_{n-1} $$

Therefore $x_n=\cfrac{1+c_{n-1}}{c_n}\,x_{n-1}\,$, then telescoping gives $x_n$ in terms of $x_1$ and $\,c_2,c_3,\cdots,c_n$.