{ ((u_0 = 3 : u_1 = 4)),((u_(n+1) = u_n + 6u_(n−1) )) :} Express u


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Question Number 168233 by alcohol last updated on 06/Apr/22
${ ((u_0 = 3 : u_1 = 4)),((u_(n+1) = u_n + 6u_(n−1) )) :} Express u_n in terms of n$
${\begin{cases} u_{0} = 3 : u_{1} = 4 \\ u_{n + 1} = u_{n} + 6 u_{n - 1} \end{cases}$ $E x p r e s s u_{n} i n t e r m s o f n$
	Answered by mr W last updated on 06/Apr/22

	$r^{2} - r - 6 = 0$ $(r + 2) (r - 3) = 0$ $r_{1} = - 2, r_{2} = 3$ $u_{n} = A \times {(- 2)}^{n} + B \times 3^{n}$ $u_{0} = A + B = 3$ $u_{1} = - 2 A + 3 B = 4$ $\Rightarrow B = 2, A = 1$ $\Rightarrow u_{n} = {(- 2)}^{n} + 2 \times 3^{n}$
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