A Sequence$b _ { 1 } , b _ { 2 } , b _ { 3 } , \ldots$

Question

Examlex · Accepted Answer

The Answer is The characteristic equation is

t ^ { 2 } - 2 t - 8 = 0

. Since

t ^ { 2 } - 2 t - 8 = ( t - 4 ) ( t + 2 )

, there are two roots:

t = 4

and

t = - 2

By the distinct roots theorem, there exist constants

C

and

D

such that

b _ { n } = C \cdot 4 ^ { n } + D \cdot ( - 2 ) ^ { n } \quad

for all integers

n \geq 0 .

Since

b _ { 0 } = 1

and

b _ { 1 } = 0

, then

\begin{array} { l } \left\{ \begin{array} { l } b _ { 0 } = C \cdot 4 ^ { 0 } + D \cdot ( - 2 ) ^ { 0 } = C + D = 1 \b _ { 1 } = C \cdot 4 ^ { 1 } + D \cdot ( - 2 ) ^ { 1 } = 4 C - 2 D = 0\end{array} ight\} \Leftrightarrow \left\{ \begin{array} { l } C = 1 - D \D = 2 C\end{array} ight\} \\Leftrightarrow \left\{ \begin{array} { l } C = 1 - 2 C \D = 2 C\end{array} ight\} \Leftrightarrow \left\{ \begin{array} { l } C = 1 / 3 \D = 2 / 3\end{array} ight\}\end{array}

Thus

b _ { n } = \frac { 1 } { 3 } \cdot 4 ^ { n } + \frac { 2 } { 3 } \cdot ( - 2 ) ^ { n }

for all integers

n \geq 0

.

A Sequence $b _ { 1 } , b _ { 2 } , b _ { 3 } , \ldots$