0
votes
1answer
60 views

Induction proof of function from $\mathbb R$ to $\mathbb R$

Let f be a function from $\mathbb R$ to $\mathbb R$ satisfying $f(\frac{x_1+x_2}{2})=\frac{f(x_1)+f(x_2)}{2}$ Prove that for any positive integer $n$ we have ...
4
votes
2answers
66 views

Fibonacci induction stuck in adding functions together

Using Fibonacci... I am Proving: $$f_3 + f_6 + \cdots + f_{3n} = \frac12(f_{3n+2}-1) $$ I did the assumption of $f_1$ which gave $\mathrm{LHS}=2=\mathrm{RHS}$. For the second part where it is $n+1$ ...
1
vote
0answers
65 views

How to solve a inductively defined set?

I'm new to induction. Right now iIm working on a task which I'm not sure if I've solved it correctly. Here is the task: Give an inductive definition of the given language below: ...
0
votes
2answers
50 views

$g(n_0) \le f(n_0) , \frac {g(n+1)}{g(n)} \geq \frac {f(n+1)}{f(n)} \implies g(n) ≤ f(n)$ (By Induction )

Let $f, g : \Bbb N \to \Bbb N$ and $n_0 \in \Bbb N$ be a natural number such that $g(n_0) \le f(n_0)$, and for each $n \ge n_0$, Then for any $n \ge n_0: g(n) ≤ f(n)$�
3
votes
1answer
98 views

A formula for n-derivative of the inverse of a function?

Let $y=f^{-1}(x)$. As we know: \begin{align} \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{1}{{f}'(y)} \end{align} Thereof we have: \begin{align} \frac{\mathrm{d^2} y}{\mathrm{d} ...
0
votes
0answers
109 views

Induction proof of surjectivity

I have a problem. Let $A: S\to T$ be a surjective map between finite sets. Prove by induction that $|S|\geq|T|$ and that if $|S|=|T|$, then $A$ is bijective. Another way to phrase the question is: ...
6
votes
4answers
608 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 $f(2) = 5$.
18
votes
5answers
594 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
125 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
84 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
143 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 ...
4
votes
3answers
364 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) = ...