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Question Number 31990 by gunawan last updated on 17/Mar/18

$$a_n=2a_{n-1}+3a_{n-2}$$$$a_0=1$$$$a_1=2$$$$a_n=...$$

Commented by mondodotto@gmail.com last updated on 18/Mar/18

$${\mathit{a}}_{\mathit{n}}=2{\mathit{a}}_{\mathit{n}-1}+3{\mathit{a}}_{\mathit{n}-2}$$$$\mathit{a}\mathit{s}\mathit{s}\mathit{t}\mathit{a}\mathit{t}\mathit{e}\mathit{d}\mathit{a}\mathit{b}\mathit{o}\mathit{v}\mathit{e}$$

Commented by Tinkutara last updated on 18/Mar/18

$$a_n=\frac{1}{4}[3^{n+1}+{(-1)}^n]$$

Commented by gunawan last updated on 18/Mar/18

$$\mathrm{thank}\mathrm{you}$$

Answered by prakash jain last updated on 18/Mar/18

$$\mathrm{Characteristic}\mathrm{Equation}$$$$x^2-2x-3=0$$$$(x-3)(x+1)=0$$$$x=3,-1$$$$a_n=c_1{\left(3\right)}^n+c_2{(-1)}^n$$$$a_0=1\Rightarrow c_1+c_2=1$$$$a_1=2\Rightarrow 3c_1-c_2=2$$$$c_1=\frac{3}{4},c_2=\frac{1}{4}$$$$a_n=\frac{{\left(3\right)}^{n+1}}{4}+\frac{{(-1)}^n}{4}$$

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