Resolve-1-u-n-2-2u-n-1-4u-n-3-n-with-u-o-1-u-1-2-2-u-n-u-n-1-u-n-2-2sin-n-3-with-u-o-1-u-1-2-

Question Number 159560 by LEKOUMA last updated on 18/Nov/21

R e s o l v e

1 . u_{n + 2} - 2 u_{n + 1} + 4 u_{n} = 3^{n}

w i t h u_{o} = 1, u_{1} = - 2

2 . u_{n} = u_{n - 1} - u_{n - 2} + 2 \sin (\frac{n Π}{3})

w i t h u_{o} = 1, u_{1} = 2

Answered by mr W last updated on 02/Dec/21

(1)

u_{n + 2} - 2 u_{n + 1} + 4 u_{n} = 3^{n}

l e t u_{n} = k \times 3^{n} + v_{n}

k 3^{n + 2} - 2 k 3^{n + 1} + 4 k 3^{n} = 3^{n}

9 k - 6 k + 4 k = 1

7 k = 1

k = \frac{1}{7}

\Rightarrow u_{n} = v_{n} + \frac{3^{n}}{7}

v_{n + 2} - 2 v_{n + 1} + 4 v_{n} = 0

r^{2} - 2 r + 4 = 0

r = 1 \pm \sqrt{3} i = 2 (\cos \frac{π}{3} \pm i \sin \frac{π}{3})

v_{n} = 2^{n + 1} {C \cos \frac{n π}{3} + D \sin \frac{n π}{3}}

v_{0} = u_{0} - \frac{1}{7} = 1 - \frac{1}{7} = \frac{6}{7}

v_{1} = u_{1} - \frac{3}{7} = - 2 - \frac{3}{7} = - \frac{17}{7}

v_{0} = 2 C = \frac{6}{7}

\Rightarrow C = \frac{6}{14}

v_{1} = 4 (C \cos \frac{π}{3} + D \sin \frac{π}{3}) = - \frac{17}{7}

2 (C + \sqrt{3} D) = - \frac{17}{7}

\Rightarrow D = - \frac{23}{14 \sqrt{3}}

v_{n} = \frac{2^{n}}{7} (6 \cos \frac{n π}{3} - \frac{23}{\sqrt{3}} \sin \frac{n π}{3})

\Rightarrow u_{n} = \frac{2^{n}}{7} (6 \cos \frac{n π}{3} - \frac{23}{\sqrt{3}} \sin \frac{n π}{3}) + \frac{3^{n}}{7}

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