0
votes
0answers
22 views

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 ...
2
votes
5answers
680 views

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 ...
0
votes
1answer
39 views

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 ...
1
vote
0answers
27 views

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 ...
1
vote
3answers
158 views

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) } } ...
2
votes
1answer
93 views

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 ...
17
votes
1answer
218 views

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: ...
2
votes
1answer
52 views

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 ...
3
votes
1answer
211 views

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 ...
0
votes
2answers
91 views

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 ...
0
votes
0answers
28 views

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 ...
1
vote
1answer
74 views

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 ...
2
votes
1answer
62 views

Recurrence relation help

The function $\psi_k(n)$ satisfies the recurrence relation: $$\sum_{j=0}^k\binom{k}{j}(-1)^j\psi_j(n)\ln(n)^{k-j}=\psi_k(n)$$ Using this, is there a general way I can re-write the function $ ...
1
vote
6answers
107 views

limit of convergent series

What is the limit of $U_{n+1} = \dfrac{2U_n + 3}{U_n + 2}$ and $U_0 = 1$? I need the detail, and another way than using the solution of $f(x)=x$, as $f(x) = \frac{2x+3}{x+2}$ because I can't show ...
3
votes
1answer
1k views

Recurrences that cannot be solved by the master theorem

I am given this problem as extra credit in my class: Propose TWO example recurrences that CANNOT be solved by the Master Theorem. Note that your examples must follow the shape that $T(n) = ...
6
votes
1answer
322 views

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: ...
0
votes
1answer
542 views

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 ...
3
votes
3answers
460 views

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 ...
3
votes
2answers
1k views

Deriving formulas for recursive functions

If I had a recursive function (f(n) = f(n-1) + 2*f(n-2) for example), how would I derive a formula to solve this? For example, with the Fibonacci sequence, Binet's ...