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Difference between revisions of "2007 iTest Problems/Problem 47"

(Created page with "== Problem == Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qqu...")
 
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== Solution ==
 
== Solution ==
 +
1447
 +
==See Also==
 +
{{iTest box|year=2007|num-b=46|num-a=48}}

Revision as of 11:23, 6 April 2024

Problem

Let $\{X_n\}$ and $\{Y_n\}$ be sequences defined as follows: $X_0=Y_0=X_1=Y_1=1$,

\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\ Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}

Let $k$ be the largest integer that satisfies all of the following conditions: $|X_i-k|\leq 2007$, for some positive integer $i$; $|Y_j-k|\leq 2007$, for some positive integer $j$; and $k<10^{2007}$. Find the remainder when $k$ is divided by $2007$.

Solution

1447

See Also

2007 iTest (Problems, Answer Key)
Preceded by:
Problem 46 		Followed by:
Problem 48
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