A pupil is writing on a board positive integers $x_0,x_1,x_2,x_3...$ after the following algorithm which implies arithmetic progression $3,5,7,9...$.Each term of rank $k\ge2$ is a difference between the product of the last number on the board and the term of arithmetic progression of rank $k$ and the last but one term on the bord with the sum of the terms of the arithemtic progression with ranks less than $k$.If $x_0=0 $ and $x_1=1$ find $x_n$ according to n.
Problem
Source: Moldova TST 2018
Tags: algorithm, arithmetic sequence, combinatorics
Jalil_Huseynov
12.11.2021 21:02
Statement can be changed to this: Let $a_n=2n+1$ and $(x_n)$ be sequnce such that $x_0=0$,$x_1=1$ and for any $n\geq 2$ $x_n=x_{n-1}\cdot a_n-x_{n-2}\cdot \sum_{i=1}^{k-1}a_i$. Then find explicit formula of $x_n$ as a function of $n$.
$x_n=x_{n-1}\cdot (2n+1)-x_{n-2}\cdot(n-1)^2$. Find explicit formula of $x_n$ as a function of $n$.
Jalil_Huseynov
12.11.2021 21:21
Any idea?
MathLuis
13.11.2021 02:12
Jalil_Huseynov wrote: Statement can be changed to this: Let $a_n=2n+1$ and $(x_n)$ be sequnce such that $x_0=0$,$x_1=1$ and for any $n\geq 2$ $x_n=x_{n-1}\cdot a_n-x_{n-2}\cdot \sum_{i=1}^{k-1}a_i$. Then find explicit formula of $x_n$ as a function of $n$.
$x_n=x_{n-1}\cdot (2n+1)-x_{n-2}\cdot(n^2-1)$. Find explicit formula of $x_n$ as a function of $n$.
U sure this is right? When i tried to find the generating function of $x_n$ is got something that involves $e$ and the exponential integral, which is kinda weird. Edit: the shorter version is $x_n=(2n+1)x_{n-1}-(n-1)^2 x_{n-2}$ for any $n \ge 2$
Jalil_Huseynov
13.11.2021 07:17
MathLuis wrote: Jalil_Huseynov wrote: Statement can be changed to this: Let $a_n=2n+1$ and $(x_n)$ be sequnce such that $x_0=0$,$x_1=1$ and for any $n\geq 2$ $x_n=x_{n-1}\cdot a_n-x_{n-2}\cdot \sum_{i=1}^{k-1}a_i$. Then find explicit formula of $x_n$ as a function of $n$.
$x_n=x_{n-1}\cdot (2n+1)-x_{n-2}\cdot(n^2-1)$. Find explicit formula of $x_n$ as a function of $n$.
U sure this is right? When i tried to find the generating function of $x_n$ is got something that involves $e$ and the exponential integral, which is kinda weird. Interesting. I found a pdf of Moldova TST 2018 and in there statement of Day1 P4 was written like that I said. I actually didn't check that are they similar or not,I just post exact statement which is given in this pdf.
MathLuis
18.12.2021 21:01
Can u share that pdf @above pls, i want to see it becuase i couldnt find anything about this problem in other part.
Jalil_Huseynov
18.12.2021 21:10
MathLuis wrote: Can u share that pdf @above pls, i want to see it becuase i couldnt find anything about this problem in other part. Okay, I will DM you.
eisenberg.k
10.03.2022 02:44
Bump!!!!