Is there any way to get the recursive formula of the form $r_n=\alpha r_{n-1}+\beta$ to single formula as a function of $n$. I've seen results that find single formula as function of $n$ for geometric series and such but can't seem to find any literature or results for this type.
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1See this. – Git Gud Oct 20 '14 at 09:17
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aren't those of the form $r_n=\alpha r_{n-1}+\beta r_{n-2} $ – Kamster Oct 20 '14 at 09:22
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Hint: If $\alpha \ne 1$, ${r_n}=\alpha {r_{n-1}}+\beta \Rightarrow {r_n}+\frac{\beta }{\alpha -1}=\alpha \left( {r_{n-1}}+\frac{\beta }{\alpha -1} \right)$.
Eclipse Sun
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I guess Im not completely seeing why this is helpful, Ive never been very good at these types of questions – Kamster Oct 20 '14 at 09:25
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@Kamster This shows that $\left { r_n + \frac {\beta} {\alpha -1} \right }$ is a geometric progression. – Eclipse Sun Oct 20 '14 at 09:28