a_(n+2) −5a_(n+1) +6a_n =3n+5^n a_1 =1, a_2 =0 find a


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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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