# Tagged Questions

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### 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 ...
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### 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 $n+1$ ...
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### 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: ...
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### 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} ...
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### 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: ...
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### $f: \mathbb{R} \to \mathbb{R}$ 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 $f(2) = 5$.
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### IMO 1987 - function

Show that there is no function $f: \mathbb{N} \to \mathbb{N}$ such that $f(f(n))=n+1987, \ \forall n \in \mathbb{N}$.
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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 $f(a,b) = ... 2answers 86 views ### how to prove${{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 ...
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) = ...