Found problems: 492
The first three terms of an arithmetic sequence are $ 2x\minus{}3$, $ 5x\minus{}11$, and $ 3x\plus{}1$ respectively. The $ n$th term of the sequence is $ 2009$. What is $ n$?
$ \textbf{(A)}\ 255 \qquad
\textbf{(B)}\ 502 \qquad
\textbf{(C)}\ 1004 \qquad
\textbf{(D)}\ 1506 \qquad
\textbf{(E)}\ 8037$
Prove that for every real number $M$ there exists an infinite arithmetic progression such that:
- each term is a positive integer and the common difference is not divisible by 10
- the sum of the digits of each term (in decimal representation) exceeds $M$.
A polynomial $p(x)$ with real coefficients is said to be [i]almeriense[/i] if it is of the form:
$$
p(x) = x^3+ax^2+bx+a
$$
And its three roots are positive real numbers in arithmetic progression. Find all [i]almeriense[/i] polynomials such that $p\left(\frac{7}{4}\right) = 0$
For every $ n$ the sum of $ n$ terms of an arithmetic progression is $ 2n \plus{} 3n^2$. The $ r$th term is:
$ \textbf{(A)}\ 3r^2 \qquad \textbf{(B)}\ 3r^2 \plus{} 2r \qquad \textbf{(C)}\ 6r \minus{} 1 \qquad \textbf{(D)}\ 5r \plus{} 5 \qquad \textbf{(E)}\ 6r \plus{} 2 \qquad$
Let $ \left( x_n\right)_{n\ge 1} $ be a sequence having $ x_1=3 $ and defined as $ x_{n+1} =\left\lfloor \sqrt 2x_n\right\rfloor , $ for every natural number $ n. $ Find all values $ m $ for which the terms $ x_m,x_{m+1},x_{m+2} $ are in arithmetic progression, where $ \lfloor\rfloor $ denotes the integer part.
How many sets of two or more consecutive positive integers have a sum of 15?
$ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$
A sequence $a_1,a_2,\ldots ,a_n,\ldots$ of natural numbers is defined by the rule
\[a_{n+1}=a_n+b_n\ (n=1,2,\ldots)\]
where $b_n$ is the last digit of $a_n$. Prove that such a sequence contains infinitely many powers of $2$ if and only if $a_1$ is not divisible by $5$.
In a football tournament there are n teams, with ${n \ge 4}$, and each pair of teams meets exactly once. Suppose that, at the end of the tournament, the final scores form an arithmetic sequence where each team scores ${1}$ more point than the following team on the scoreboard. Determine the maximum possible score of the lowest scoring team, assuming usual scoring for football games (where the winner of a game gets ${3}$ points, the loser ${0}$ points, and if there is a tie both teams get ${1}$ point).
The sequence \[\log_{12}{162},\, \log_{12}{x},\, \log_{12}{y},\, \log_{12}{z},\, \log_{12}{1250}\] is an arithmetic progression. What is $x$?
$ \textbf{(A)} \ 125\sqrt{3} \qquad \textbf{(B)} \ 270 \qquad \textbf{(C)} \ 162\sqrt{5} \qquad \textbf{(D)} \ 434 \qquad \textbf{(E)} \ 225\sqrt{6}$
Prove that if three prime numbers form an arithmetic progression whose difference is not divisible by 6, then the smallest of these numbers is $3 $.
9.7 Is there an infinite arithmetic sequence $\{a_n\}\subset \mathbb N$ s.t. $a_n+...+a_{n+9}\mid a_n...a_{n+9}$ for all $n$?
([i]V. Senderov[/i])
From an infinite arithmetic progression $ 1,a_1,a_2,\dots$ of real numbers some terms are deleted, obtaining an infinite geometric progression $ 1,b_1,b_2,\dots$ whose ratio is $ q$. Find all the possible values of $ q$.
Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.
Arithmetic sequence with all items real, and the common difference is $4$. If the sum of the square of the first item and all items else is not more than $100$, then there are________items at most.
There exists a unique strictly increasing arithmetic sequence $\{a_i\}_{i=1}^{100}$ of positive integers such that \[a_1+a_4+a_9+\cdots+a_{100}=\text{1000},\] where the summation runs over all terms of the form $a_{i^2}$ for $1\leq i\leq 10$. Find $a_{50}$.
[i]Proposed by David Altizio and Tony Kim[/i]
Let $a, \overline{bcd}, \overline{aef}, \overline{cfg}, \overline{hci}, \overline{dea}, \overline{ifd}, \overline{jgf}, \overline{bfeg},\ldots$ be an increasing arithmetic progression. Find the $16$th term of this sequence.
Prove that for any positive integer $ k$, there exists an arithmetic sequence $ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \frac{a_3}{b_3}, ... ,\frac{a_k}{b_k}$ of rational numbers, where $ a_i, b_i$ are relatively prime positive integers for each $ i \equal{} 1,2,...,k$ such that the positive integers $ a_1, b_1, a_2, b_2, ..., a_k, b_k$ are all distinct.
Given a positive integer $a$, prove that $\sigma(am) < \sigma(am + 1)$ for infinitely many positive integers $m$. (Here $\sigma(n)$ is the sum of all positive divisors of the positive integer number $n$.)
[i]Vlad Matei[/i]
Let $a_1, a_2, ..., a_k$ be a finite arithmetic sequence with
\[ a_4+a_7+a_{10}=17 \] and \[ a_4+a_5+a_6+a_7+a_8+a_9+a_{10}+a_{11}+a_{12}+a_{13}+a_{14}=77 \] If $a_k=13$, then $k=$
$ \textbf{(A)}\ 16 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 22 \qquad\textbf{(E)}\ 24 $
The three sides of a right triangle have integral lengths which form an arithmetic progression. One of the sides could have length
$\text{(A)}\ 22 \qquad \text{(B)}\ 58 \qquad \text{(C)}\ 81 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 361$
Let $n$ be a positive integer and let $x_1\le x_2\le\cdots\le x_n$ be real numbers.
Prove that
\[
\left(\sum_{i,j=1}^{n}|x_i-x_j|\right)^2\le\frac{2(n^2-1)}{3}\sum_{i,j=1}^{n}(x_i-x_j)^2.
\]
Show that the equality holds if and only if $x_1, \ldots, x_n$ is an arithmetic sequence.
Let be a bipartition of the set formed by the first $ 13 $ nonnegative numbers. Prove that at least one of these two subsets that form this partition contains an arithmetic progression.
$A_1, A_2,\dots A_k$ are different subsets of the set $\{1,2,\dots ,2016\}$. If $A_i\cap A_j$ forms an arithmetic sequence for all $1\le i <j\le k$, what is the maximum value of $k$?
The roots of the equation
\[ x^3-3ax^2+bx+18c=0 \]
form a non-constant arithmetic progression and the roots of the equation
\[ x^3+bx^2+x-c^3=0 \]
form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.
Two real number sequences are guiven, one arithmetic $\left(a_n\right)_{n\in \mathbb {N}}$ and another geometric sequence $\left(g_n\right)_{n\in \mathbb {N}}$ none of them constant. Those sequences verifies $a_1=g_1\neq 0$, $a_2=g_2$ and $a_{10}=g_3$. Find with proof that, for every positive integer $p$, there is a positive integer $m$, such that $g_p=a_m$.