Tagged Questions
6
votes
4answers
287 views
$f: \mathbb{R} \to \mathbb{R}$ satisfies $(x-2)f(x)-(x+1)f(x-1) = 3$. Evaluate $f(2013)$, given that $f(2)=5$
The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $(x-2)f(x)-(x+1)f(x-1) = 3$. Evaluate $f(2013)$, given that $f(2) = 5$.
15
votes
5answers
465 views
IMO 1987 - function
Show that there is no function $f: \mathbb{N} \to \mathbb{N}$ such that $f(f(n))=n+1987, \ \forall n \in \mathbb{N}$.
1
vote
2answers
86 views
Tricky well defined function and induction
Lets define a function $f$ such that $\Bbb N \times\Bbb N \to\Bbb N$.
It takes two natural numbers as inputs and also outputs a natural number.
Let $f$ have the following properties
$f(a,b) = ...
0
votes
2answers
75 views
how to prove ${{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+\cdots +{{a}_{n}}{{x}^{n}}=0$ has at least one real root in $(0,1)$. [duplicate]
Possible Duplicate:
Prove existence of a real root.
If $a_0$+$\frac{a_1}{2}$+$\frac{a_2}{3}+\ldots+\frac{a_n}{n+1}=0$, how to prove ${{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+\cdots ...
0
votes
3answers
78 views
Induction on functions
I'm working through some homework on induction, and most problems I can solve fine, but I have problem getting started on induction proofs that ask you to prove function relations. For example, here ...
1
vote
0answers
68 views
Using induction to prove $f(n)=2n-3$ if $n\lt4$, $f(n)=2n-4$ if $n\ge4$
I have a problem, for which the solution(by looking at the pattern) I found is
$$f(n)=\begin{cases}2n-3,\text{if }n<4\\2n-4,\text{if }n\ge 4\;.\end{cases}$$
I want to prove it inductively, I'm ...
4
votes
3answers
344 views
Question about a recursively defined function
Problem. Let $(f_n)_{n=1}^\infty$ be a sequence of functions $f_n\colon [-1,\infty)^n\to\mathbb{R}$ that are recursively defined in the following way:
$$f_1(x_1)=1+x_1,$$
$$f_n(x_1,\ldots,x_n) = ...
