User Michael Hartley

8 votes

Accepted

If $a$ is a root of $x^2+ x + 1$, simplify $1 + a + a^2 +\dots+ a^{2017}.$

answered Aug 28, 2017 at 1:38

6 votes

Accepted

Fibonacci 19^n is multiple of 19 a mathematical property or a unique event?

answered Nov 24, 2022 at 7:09

4 votes

The sequence $(x_n)$ : $x_{n+2}=\frac{x_{n+1}\sqrt{x_n^2+1}+x_{n}\sqrt{x_{n+1}^2+1}-x_n-x_{n+1}}{x_{n+1}x_n-(\sqrt{x_{n+1}^2+1}-1)(\sqrt{x_n^2+1}-1)}$

answered Aug 30, 2021 at 8:40

4 votes

Accepted

Write all permutations in the cyclic group $C_{12}$ of order 12 in cycle notation.

answered Nov 16, 2021 at 5:30

4 votes

Accepted

Let $V$ be the space spanned by $v_1=\cos^2x$, $v_2=\sin^2x$ and $v_3=\cos2x$.

answered Aug 28, 2017 at 1:19

4 votes

If $450^\circ<\alpha<540^\circ$ and $\cot\alpha=-\frac{7}{24},$ calculate $\cos\frac{\alpha}{2}$. Why is my solution wrong?

answered Aug 14, 2017 at 8:07

4 votes

Accepted

Showing that either $v$ is an eigenvector for $2\times 2$ matrix $A$ or else $(A − \lambda \operatorname{Id})v$ is an eigenvector for $A$.

answered Feb 5, 2019 at 2:17

4 votes

If I flip two fair coins, and then tell you that one is heads, what is the probability that the other coin is also heads?

answered Feb 5, 2019 at 2:22

4 votes

A Number Theoretic Game

answered Jun 23, 2021 at 6:55

4 votes

Accepted

Proving the sum to product formula with complex numbers starting from the left hand side

answered Aug 30, 2021 at 1:13

3 votes

Accepted

What is the maximum number of distinct integers $A$ can have as elements?

answered Jun 23, 2021 at 8:30

3 votes

Accepted

Relation of surface area of a sphere to its volume.

answered Jan 14, 2021 at 0:47

3 votes

$P(X^2+Y^2<1)$ when $X,Y\sim U[0,1]$

answered Dec 11, 2017 at 1:59

3 votes

Coloring a rectangle with 3 rows and 4 columns using two colors.

answered Aug 23, 2017 at 7:39

3 votes

Accepted

$a$ algebraic number, $\exists$ integer $n>0$ such that $na$ algebraic integer

answered Sep 19, 2017 at 5:30

3 votes

Finding Limit of an Integral: $\lim_{n\to\infty}\int_a^b f(x)\sin^3{(nx)} \:dx$

answered Jul 31, 2017 at 3:04

3 votes

Accepted

The number of real roots of an exponential equation with 4 parameters

answered Nov 16, 2021 at 6:58

3 votes

Powers of $2$ starting with $123$...Does a pattern exist?

answered Apr 8, 2021 at 9:25

2 votes

Accepted

Prove that $\theta \cdot (x,y)=(x\cos(\theta),x\sin(\theta)+y\cos(\theta) )$ is an group action.

answered Dec 7, 2022 at 1:32

2 votes

Accepted

What is the largest number of elements of $\{\sin\alpha\cos\beta,\sin\beta\cos\gamma,\sin\gamma\cos\alpha\}$ (all angles acute) that can exceed $1/2$?

answered Sep 3, 2021 at 1:47

2 votes

Accepted

Prove that $\sum_{i=0}^{m}{\binom{n-i}{k}}=\binom{n+1}{k+1}-\binom{n-m}{k+1}$

answered Sep 6, 2021 at 1:05

2 votes

Accepted

Probability Distribution Calculation of Variables

answered Oct 3, 2022 at 3:58

2 votes

Even integer has at least $K$ representations as the sum of two primes for any $K$

answered Nov 23, 2022 at 4:45

2 votes

Accepted

Help crafting equation - How to calculate all invalid combinations

answered Nov 23, 2022 at 5:10

2 votes

I have a mean and a desired range for a normal distribution, how do I discover what standard deviation I need?

answered Sep 25, 2013 at 3:29

2 votes

Rational numbers raised to an irrational power.

answered Aug 1, 2017 at 7:54

2 votes

solving this linear equation by substitution?

answered Aug 8, 2017 at 7:36

2 votes

Induction on Fermat Numbers: $F_n = \prod_{j=0}^{n-1}F_j+2$

answered Aug 9, 2017 at 0:36

2 votes

Accepted

How to convert $r(t,s)=\langle t+s,t-s,t^2+s^2\rangle$ to Cartesian coordinates?

answered Aug 14, 2017 at 5:56

2 votes

How to solve this type of permutations product question

answered Dec 4, 2017 at 14:00

Stack Exchange Network

98 Answers

If $a$ is a root of $x^2+ x + 1$, simplify $1 + a + a^2 +\dots+ a^{2017}.$

Fibonacci 19^n is multiple of 19 a mathematical property or a unique event?

The sequence $(x_n)$ : $x_{n+2}=\frac{x_{n+1}\sqrt{x_n^2+1}+x_{n}\sqrt{x_{n+1}^2+1}-x_n-x_{n+1}}{x_{n+1}x_n-(\sqrt{x_{n+1}^2+1}-1)(\sqrt{x_n^2+1}-1)}$

Write all permutations in the cyclic group $C_{12}$ of order 12 in cycle notation.

Let $V$ be the space spanned by $v_1=\cos^2x$, $v_2=\sin^2x$ and $v_3=\cos2x$.

If $450^\circ<\alpha<540^\circ$ and $\cot\alpha=-\frac{7}{24},$ calculate $\cos\frac{\alpha}{2}$. Why is my solution wrong?

Showing that either $v$ is an eigenvector for $2\times 2$ matrix $A$ or else $(A − \lambda \operatorname{Id})v$ is an eigenvector for $A$.

If I flip two fair coins, and then tell you that one is heads, what is the probability that the other coin is also heads?

A Number Theoretic Game

Proving the sum to product formula with complex numbers starting from the left hand side

What is the maximum number of distinct integers $A$ can have as elements?

Relation of surface area of a sphere to its volume.

$P(X^2+Y^2<1)$ when $X,Y\sim U[0,1]$

Coloring a rectangle with 3 rows and 4 columns using two colors.

$a$ algebraic number, $\exists$ integer $n>0$ such that $na$ algebraic integer

Finding Limit of an Integral: $\lim_{n\to\infty}\int_a^b f(x)\sin^3{(nx)} \:dx$

The number of real roots of an exponential equation with 4 parameters

Powers of $2$ starting with $123$...Does a pattern exist?

Prove that $\theta \cdot (x,y)=(x\cos(\theta),x\sin(\theta)+y\cos(\theta) )$ is an group action.

What is the largest number of elements of $\{\sin\alpha\cos\beta,\sin\beta\cos\gamma,\sin\gamma\cos\alpha\}$ (all angles acute) that can exceed $1/2$?

Prove that $\sum_{i=0}^{m}{\binom{n-i}{k}}=\binom{n+1}{k+1}-\binom{n-m}{k+1}$

Probability Distribution Calculation of Variables

Even integer has at least $K$ representations as the sum of two primes for any $K$

Help crafting equation - How to calculate all invalid combinations

I have a mean and a desired range for a normal distribution, how do I discover what standard deviation I need?

Rational numbers raised to an irrational power.

solving this linear equation by substitution?

Induction on Fermat Numbers: $F_n = \prod_{j=0}^{n-1}F_j+2$

How to convert $r(t,s)=\langle t+s,t-s,t^2+s^2\rangle$ to Cartesian coordinates?

How to solve this type of permutations product question