Articles of arithmetic progressions

Primes and arithmetic progressions

In a book on complex analysis, the authors prove: Given finitely many (non-trivial) arithmetic progressions of natural numbers $$a_1, a_1+d_1, a_1+2d_1, \cdots $$ $$a_2, a_2+d_2, a_2+2d_2, \cdots, $$ $$a_k, a_k+d_k, a_k+2d_k, \cdots, $$ their totality (union) is never $\mathbb{N}$. Given: any $k$ (non-trivial) arithmetic progressions. Let $S$ denote the set of those numbers which are […]

Sum of first n natural numbers proof

I know how to prove this by induction but the text I’m following shows another way to prove it and I guess this way is used again in the future. I’m confused by it. So the expression for first n numbers is: $$\frac{n(n+1)}{2}$$ And this second proof starts out like this. It says since: $$(n+1)^2-n^2=2n+1$$ […]

If $a+b+c=3$, find the greatest value of $a^2b^3c^2$.

If $a+b+c=3$, and $a,b,c>0$ find the greatest value of $a^2b^3c^2$. I have no idea as to how I can solve this question. I only require a small hint to start this question. It would be great if someone could help me with this.

Cube of harmonic mean

Question: The geometric mean of two numbers is $8$ while the arithmetic mean is $4$. Determine the cube of the harmonic mean. Answer is $4096$. Can anyone tell me how to solve this problem? I do not know how since from what I’ve known, the AM of is always greater than GM. Please show me […]

Are there arbitrarily large sets $S$ of natural integers such that the difference of each pair is their GCD?

I am interested in sets $S \subseteq \mathbb{N}$ of natural integers with the following property: for any $i, j \in S$, then the greatest common divisor of $i$ and $j$ is the absolute value of their difference, $|i – j|$. Equivalently, this amounts to requiring that the difference $|i-j|$ divides $i$ and $j$. Are there […]

Is there a closed form of the sum $\sum _{n=1}^x\lfloor n \sqrt{2}\rfloor$

I need to find the n-th partial sum of: $$\sum _{n=1}^x\lfloor n \sqrt{2}\rfloor$$ Or this sum of a Beatty sequence. I tried to expand as the following: $$=\frac{\sqrt{2} x \left(x+1\right)}{2}-\frac{x}{2}+\frac{1}{\pi }\sum _{n=1}^x\sum _{k=0}^{\infty}\frac{\sin \left(2 \pi \sqrt{2} k\ n\right)}{k}$$ $$=\frac{\sqrt{2} x \left(x+1\right)}{2}-\frac{x}{2}+\frac{1}{\pi }\sum _{n=1}^x \arctan (\tan( \frac{\pi – 2 \pi \sqrt{2} n}{2}))$$ For $ n ∈ […]

Is there a primitive recursive function which gives the nth digit of $\pi$, despite the table-maker's dilemma?

I asked a question about this a while ago and it got deleted, so I’ve looked into it a bit more and I’ll explain my problem better. Planetmath.org told me that there is a primitive recursive function which gives the nth digit of $\pi$, but didn’t prove it. When I asked before how one might […]

$S_n = x_1^3+x_2^3+ \cdots +x_n^3$ squares perfect

It is considered arithmetic progression $x_1,x_2, \cdots, x_n,\cdots, x_1 \neq0$ . Show that if sums $$S_n = x_1^3+x_2^3+ \cdots +x_n^3$$ is squares perfect for any natural $n \in N$, then there are $k\in N^*$ so $x_n=nk^2$, for any $n \in N$. All my attempts were fruitless.

How to find a general sum formula for the series: 5+55+555+5555+…?

I have a question about finding the sum formula of n-th terms. Here’s the series: $5+55+555+5555$+…… What is the general formula to find the sum of n-th terms? My attempts: I think I need to separate 5 from this series such that: $5(1+11+111+1111+….)$ Then, I think I need to make the statement in the parentheses […]

How can I solve this question on Quadratic Equations and Arithmetic Progression?

This question already has an answer here: A.P. terms in a Quadratic equation. 2 answers