# Tagged Questions

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

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### Solving differential recurrence equations

I played around trying to make an equation describing Fibonacci numbers and ended up finding out that what I'd created was something called a recurrence equation: $f(x)=f(x-1)+f(x-2)$ ($f(x)$ is ...
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### Limit of $x_n^3/n^2$ when $x_{n+1}=x_n+ 1/\sqrt {x_n}$ with $x_0 \gt 0$

Let $(x_n)_{n \ge 0}$ a sequence of real numbers with $x_0 \gt 0$ and $x_{n+1}=x_n+ \frac {1}{\sqrt {x_n}}$. Check the existence and find $$L=\lim_{n \rightarrow \infty} \frac {x_n^3} {n^2}$$ ...
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### Finding the limits to sequences given by recursive transformation of a vector: $a_n = a_{n-1}M$

I'm interested in finding limits to 'recursive vector sequences' of the form $a_n = a_{n-1}M$ (where $a$ is a vector and the matrix $M$ is a transformation of $a$) but I don't know where to read about ...
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### How to solve this recurrence relation $f(n) = \frac{10+7f(n-1)}{2+f(n-1)}$

I had a problem in which I ended up getting the following recurrence relation: \begin{align} &f(1) = 7\\ &f(n) = \dfrac{10+7f(n-1)}{2+f(n-1)} \end{align} I haven't solved much recurrence ...
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### Linear recurrent sequences and matrices.

Let $k$ be a field, let $d$ be an integer greater than $1$, let $(v,x)\in k^d\times k^d$ and let $A\in k^{d\times d}$ be invertible. For all $n\in\mathbb{N}$, let define the following element of $k$:...
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### Finding a general solution of recurrences

I am unsure how to even start the questions :S I need to learn this stuff for the final exam of my subject and its hard to find a tutorial on how to answer this type of question.
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### recursive generating functions

\begin{align} f(0) & = 1 \\ f(1) & = 1 \\ f(2) & = 2 \\ f(2n) & = f(n)+f(n+1), \;\;\;n\gt1 \\ f(2n+1) & = f(n-1)+f(n), \;\;\;n\ge1 \\ \end{align} I am trying to figure out ...