Tagged Questions

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 $f(2) = 5$.
5answers
465 views

IMO 1987 - function

Show that there is no function $f: \mathbb{N} \to \mathbb{N}$ such that $f(f(n))=n+1987, \ \forall n \in \mathbb{N}$.
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) = ...
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 ...
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 ...
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 ...
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) = ...
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