# Tagged Questions

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### Functional Relationship Question on Analytic Geometry

I am solving some problems on analytic geometry. I have a set of points $\{P_1,P_2,P_3,...,P_k\}$ from wich $P_1,P_2$ are known. The rest have coordinates $P_n\big(x_n,y_n\big)$ and for any value of ...
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### Explanation of recursive function

Given is a function $f(n)$ with: $f(0) = 0$ $f(1) = 1$ $f(n) = 3f(n-1) + 2f(n-2)$ $\forall n≥2$ I was wondering if there's also a non-recursive way to describe the same function. WolframAlpha tells ...
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### Hi guys, can anyone help with this recurrence relation problem?

I'm going through practice questions for my exams but this question has left me confused: The Bessel functions of integer order, Jn(x), are described by the generating function: Derive the ...
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### Deriving recurrence relations, very stuck!

Going through past papers for my exams and cannot figure this one out, does anyone know how to do these? The Bessel functions of integer order, Jn(x), are described by a generating function of the ...
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### Recurrence relations (Big-O notation)

Say I'm given a recursive function such as: function(n) { if (n <= 1) return; int i; for(i = 0; i < n; i++) { function(0.8n) } } ...
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### How find this function $f(2^m)=?$

show that:there exists unique function $f:N^{+}\to N^{+}$,such $$f(1)=f(2)=1$$ and $$f(n)=f(f(n-1))+f(n-f(n-1))),n=3,4,\cdots$$ and Find $f(2^m),m>2,m\in N$ My try: let $n=3$ ,then we have ...
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### Show that $\lim\limits_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n^2}$ exists and is independent of the choice of $a$

Suppose $f:\mathbb{R}\to\mathbb{R}$ has period 1, and for some $q\in(0,1)$: $$|f(x)-f(y)|\leq q|x-y|\quad \forall x,y$$ Let $g(x)=x+f(x)$, for any $a\in\mathbb{R}$,define the following sequence: ...
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### Justifying onto function properties

For $m,n\ge0$ let $O(m,n)$ be the number of onto functions a) Explain why $O(m,n)=0$ when $m\lt n$ I said: since O is an onto function it implies that for all elements of n there is atleast one m ...
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### Minimum period of function such that $f\left(x+\frac{13}{42}\right)+f(x)=f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right)$

Let $f$ be a function from the set of real numbers $\mathbb{R}$ into itself such for all $x \in \mathbb{R},$ we have $|f(x)| \leq 1,f(x)\neq constant$ and ...
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### Recursive function into non-recursive

I have to express $a_n$ in terms of $n$. How do I convert this recursive function into a non-recursive one? Is there any methodology to follow in order to do the same with any recursively defined ...
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### Is this equation on the right form?

Lets assume there is a list $l$, where its items are denoted as $[a_0,...,a_n]$ and where we only consider the first and last third without the elements in b/n and while doing it recursively until we ...
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### Recurrence Equation with Polynomial Coefficients

As inspired by this question on the problem site Brilliant, Let $F_n(a,b,c)=a(b-c)^n+b(c-a)^n+c(a-b)^n$ Is it possible to obtain $F_n$ in terms of $F_3, F_2$? My attempt at a solution is as ...
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### Proving or disproving $f(n)-f(n-1)\le n, \forall n \gt 1$, for a recursive function with floors.

The Olympiad-style question I was given was as follows: A function $f:\mathbb{N}\to\mathbb{N}$ is defined by $f(1)=1$ and for $n>1$, by: ...
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### Linear homogeneous recurrence relation with constant coefficients: How does one determine the solution set?

According to my textbook and this Wikipedia article, a recurrence relation of the form $$b_0 a_n + b_1 a_{n-1} + \cdots + b_k a_{n-k} = 0$$ (EDIT: where $b_0 \neq 0$) has the following set of ...
### expansion of $\cos^k(\theta)$
Does any body know a expansion of : $\cos^k(\theta)$ in function of $\cos$ and/or $\sin$ but without power? For example : $\cos^2(\theta)=\frac{1}{2}(\cos(2\theta)+1)$, but i would want a ...