# Tagged Questions

Questions regarding functions defined recursively, such as the Fibonacci sequence.

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### Recurrence relation solution $a_{n}=\frac{a_{n-1}}{a_{n-1}-a_{n-2}}$

I want to find the analytic form of the recurrence relation $$a_{n}=\frac{a_{n-1}}{a_{n-1}-a_{n-2}},\,a_{1}=2,\,a_{2}=1$$ but when looking at the results they seem chaotic. Is it possible that it ...
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### I need a common term for a recursive sequence

Some days ago, I made a question here about a common term for recursive sequence. I gained very good solutions. Thanks again. Today, I was thinking of a general case which is given below. Assume $p$ ...
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### Solving $a_n = a_{n-1} + 7n$ for $n\ge1$ and $a_0 = 4$

First, I found the homogeneous solution: $$r^n - r^{n-1} = 0$$ $$\Rightarrow r = 1$$ So the homogeneous solution is of the form: $$c(1)^n = c$$ Then, to find a particular solution, I "guessed" the ...
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### Linear combination of sequence related to Fibonacci

Let $a= \frac{1+\sqrt{5}}{2}$ and $b= \frac{1-\sqrt{5}}{2}$. Prove that for all $c,d \in \mathbb{N}$ that $g_n = ca^n + db^n$ satisfies $g_{n+2} = g_{n+1} + g_{n}$ for all $n\in \mathbb{N}$. I'm ...
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### Lucas recurrence relation

Lucas recurrence relation is defined as: $V_n = PV_{n-1} – QV_{n-2}$; for $V_0 = 2; V_1 = P$ Here $P$ is positive integer and $Q = {-1, 1}$ or may be $+1$ or $-1$ A Fibonacci Pseudo prime with ...
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### Recurrence relations - walk on a graph

given the following undirected graph: I need to find a recurrence relation that describes the number of possible walks starting at point A. Well, naive me Iv'e defined $a_n$ and tried to find ...
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### Very confused by recurrence relations as a concept

We've been introduced to recurrence relations as a concept in my Discrete class. One question asks: Given the recurrence: $S(1) = 1$, $S(n) = 2S(n − 1) + 3 (for \ n > 1)$ prove ...
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### Find the $m$ such $a_{n+1}=a^5_{n}+487$ [closed]

Let $\{a_{n}\}$ be a sequence of positive integers, and suppose $a_{0}=m$. Further, $\{a_{n}\}$ satisfies $$a_{n+1}=a^5_{n}+487.$$ Find $m$ so that this sequence consists of square numbers for as long ...