Answer to Question #222123 in Discrete Mathematics for zoro

Question

Given the following recurrence relation (M).

an = −4an−1 + 5an−2, a0 = 2, a1 = 8

The solution of (M) is:

a. an = 3 − (−5)

n

b. an = 3 + (5)

n

c. an = (3)

n − 5

d. None of these

Answer 1

Characteristic equation:

${k^2} + 4k - 5 = 0$

$D = 16 + 20 = 36$

${k_1} = \frac{{ - 4 - 6}}{2} = - 5$

${k_2} = \frac{{ - 4 + 6}}{2} = 1$

Then

${a_n} = {C_1} \cdot {\left( { - 5} \right)^n} + {C_2} \cdot {1^n} = {C_1} \cdot {\left( { - 5} \right)^n} + {C_2}$

${a_0} = 2,\,{a_1} = 8 \Rightarrow \left\{ {\begin{matrix} {{C_1} + {C_2} = 2}\\ { - 5{C_1} + {C_2} = 8} \end{matrix}} \right. \Rightarrow {C_1} = - 1,\,{C_2} = 3$

Then

${a_n} = - 1 \cdot {\left( { - 5} \right)^n} + 3 = 3 - {\left( { - 5} \right)^n}$

Answer:a. ${a_n} = 3 - {\left( { - 5} \right)^n}$