a-n-2-5a-n-1-6a-n-3n-5-n-a-1-1-a-2-0-find-a-n-

Question Number 176746 by mr W last updated on 26/Sep/22

a_{n + 2} - 5 a_{n + 1} + 6 a_{n} = 3 n + 5^{n}

a_{1} = 1, a_{2} = 0

f i n d a_{n}

Answered by FongXD last updated on 26/Sep/22

Commented by Tawa11 last updated on 26/Sep/22

Great sirs

Answered by FongXD last updated on 26/Sep/22

Answered by FongXD last updated on 26/Sep/22

Answered by FongXD last updated on 26/Sep/22

Commented by mr W last updated on 26/Sep/22

t h a n k s s i r!

Answered by mr W last updated on 26/Sep/22

l e t a_{n} = b_{n} + a + b n + c 5^{n}

[b_{n + 2} - 5 b_{n + 1} + 6 b_{n}] + [a + b (n + 2) - 5 a - 5 b (n + 1) + 6 a + 6 b n] + [c 5^{n + 2} - 5 c 5^{n + 1} + 6 c 5^{n}] = 3 n + 5^{n}

[b_{n + 2} - 5 b_{n + 1} + 6 b_{n}] + [2 a - 3 b + 2 b n] + [6 c 5^{n}] = 3 n + 5^{n}

6 c = 1 \Rightarrow c = \frac{1}{6}

2 b = 3 \Rightarrow b = \frac{3}{2}

2 a - 3 b = 0 \Rightarrow a = \frac{3 b}{2} = \frac{9}{4}

b_{n + 2} - 5 b_{n + 1} + 6 b_{n} = 0

r^{2} - 5 r + 6 = 0

r = 2, 3

b_{n} = A 2^{n} + B 3^{n}

\Rightarrow a_{n} = A 2^{n} + B 3^{n} + \frac{9}{4} + \frac{3 n}{2} + \frac{5^{n}}{6}

a_{1} = A 2 + B 3 + \frac{9}{4} + \frac{3}{2} + \frac{5}{6} = 1

\Rightarrow 2 A + 3 B = - \frac{43}{12}

a_{2} = A 4 + B 9 + \frac{9}{4} + \frac{6}{2} + \frac{25}{6} = 0

\Rightarrow 4 A + 9 B = - \frac{113}{12}

\Rightarrow A = - \frac{2}{3}

\Rightarrow B = - \frac{3}{4}

\Rightarrow a_{n} = - \frac{2^{n + 1}}{3} - \frac{3^{n + 1}}{4} + \frac{9}{4} + \frac{3 n}{2} + \frac{5^{n}}{6}

Commented by Tawa11 last updated on 26/Sep/22

Great sir

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