MathDB

2 sequences

Source: Moldavian MO 2006

March 19, 2006

limitlinear algebramatrixvectoralgebra proposedalgebra

Problem Statement

Sequences

(x_n)_{n\ge1}

(y_n)_{n\ge1}

satisfy the relations

x_n=4x_{n-1}+3y_{n-1}

and

y_n=2x_{n-1}+3y_{n-1}

for

n\ge1

. If

x_1=y_1=5

find

x_n

and

y_n

. Calculate

\lim_{n\rightarrow\infty}\frac{x_n}{y_n}