Intereting Posts

Elliptic integrals with parameter outside $0<m<1$
Does $\bigcap_{n=1}^{+\infty}(-\frac{1}{n},\frac{1}{n}) = \varnothing$?
is the hilbert polynomial integer-valued everywhere?
Finding Eigenvectors with repeated Eigenvalues
How to prove the second mean value theorem for definite integrals
Can Peirce's Law be proven without contradiction?
Are eigenspaces and minimal polynomials sufficient for similarity?
Why $\kappa^{<\kappa}=2^{<\kappa}$, if $\kappa$ is a regular and limit cardinal?
Square root of a complex number
Discrete probability problem: what is the probability that you will win the game?
If A is a subset of B, then the closure of A is contained in the closure of B.
Mathematical Telescoping
Using Ratio test/Comparison test
Meaning of “kernel”
Could this approximation be made simpler ? Solve $n!=a^n 10^k$

Suppose that $0\neq x\in\mathbb{R}$ and $x + \frac1x\in\mathbb{Z}$. Prove that, for all $n\ge1$, $x^n + \frac1{x^n}\in\mathbb{Z}$.

I can’t figure out and understand the question. Can you give me some hints ?

- Showing that a root $x_0$ of a polynomial is bounded by $|x_0|<(n+1)\cdot c_{\rm max}/c_1$
- Choosing numbers without consecutive numbers.
- Looking for induction problems that are not formula-based
- Let $A$ be any uncountable set, and let $B$ be a countable subset of $A$. Prove that the cardinality of $A = A - B $
- Help with combinatorial proof of identity: $\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$
- Solution to a linear recurrence

- Prove $\sum_{i=1}^n i! \cdot i = (n+1)! - 1$?
- Proof by Induction for inequality, $\sum_{k=1}^nk^{-2}\lt2-(1/n)$
- the purpose of induction
- Multiplication of Set Discrete math
- discrete math>Recurrence relation>how find the general function of $a(n)=2a(n-1)+n^2$
- Show that for any subset $C\subseteq Y$, one has $f^{-1}(Y\setminus C) = X \setminus f^{-1}(C)$
- Prove this recurrence relation? (catalan numbers)
- Proving $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.
- Proving $\sum\limits_{i=0}^n i 2^{i-1} = (n+1) 2^n - 1$ by induction
- How to prove a total order has a unique minimal element

The base case of $n=1$ is true, and suppose it holds for all $k<n$ in order to do the induction step. Then

$$(x^{n-1}+1/x^{n-1})(x+1/x)=x^n+1/x^{n-2}+x^{n-2}+1/x^n=(x^n+1/x^n)+(x^{n-2}+1/x^{n-2})$$

so

$$x^n+1/x^n=(x^{n-1}+1/x^{n-1})(x+1/x)-(x^{n-2}+1/x^{n-2})$$

which is an integer, so the result follows by induction.

Hint: Expand $(x+\frac{1}{x})^n$ using the binomial theorem. Show that this is a linear combination of elements of the form $x^m+\frac{1}{x^m}$ for $m \leq n$ and of $1$. Then use induction.

Hint let $$a_{n}=x^n+\dfrac{1}{x^n}$$ then we have

$$a_{n+2}=(x+\dfrac{1}{x})a_{n+1}-a_{n}=ka_{n+1}-a_{n},x+\dfrac{1}{x}=k\in Z$$

since $a_{1}=k\in Z,a_{2}=k^2-2\in Z$

and it is by use Mathematical induction

- If $X$ is an order topology and $Y \subset X$ is closed, do the subspace topology and order topology on $Y$ coincide?
- Filtration of stopping time equal to the natural filtration of the stopped process
- Understanding the proof of $|ST||S\cap T| = |S||T|$ where $S, T$ are subgroups of a finite group
- Irreducible polynomial.
- Is an integer uniquely determined by its multiplicative order mod every prime
- $M/\Gamma$ is orientable iff the elements of $\Gamma$ are orientation-preserving
- Find all the numbers $n$ such that $\frac{12n-6}{10n-3}$ can't be reduced.
- Should this “definition” of set equality be an axiom?
- $\mathrm{lcm}(1, 2, 3, \ldots, n)$?
- How to prove or statements
- When is $CaCl(X) \to Pic(X)$ surjective?
- Advice for Self-Study
- Finding Sylow 2-subgroups of the dihedral group $D_n$
- Existence proof of the tensor product using the Adjoint functor theorem.
- Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}\mathrm dx$