a-n-2a-n-1-3a-n-2-a-0-1-a-1-2-a-n-

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

a_{n} = 2 a_{n - 1} + 3 a_{n - 2}

a_{0} = 1

a_{1} = 2

a_{n} = \dots

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

a_{n} = 2 a_{n - 1} + 3 a_{n - 2}

a s s t a t e d a b o v 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

thank you

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

Characteristic Equation

x^{2} - 2 x - 3 = 0

(x - 3) (x + 1) = 0

x = 3, - 1

a_{n} = c_{1} {(3)}^{n} + c_{2} {(- 1)}^{n}

a_{0} = 1 \Rightarrow c_{1} + c_{2} = 1

a_{1} = 2 \Rightarrow 3 c_{1} - c_{2} = 2

c_{1} = \frac{3}{4}, c_{2} = \frac{1}{4}

a_{n} = \frac{{(3)}^{n + 1}}{4} + \frac{{(- 1)}^{n}}{4}