Given-a-n-1-5a-n-6a-n-1-If-a-1-10-and-a-2-26-find-a-n-

Question Number 104544 by john santu last updated on 22/Jul/20

G i v e n a_{n + 1} = 5 a_{n} - 6 a_{n - 1}

. I f a_{1} = 10 a n d a_{2} = 26, f i n d a_{n} ?

Answered by bobhans last updated on 22/Jul/20

l e t a_{n} = A ρ^{n} \Rightarrow A ρ^{n + 1} - 5 A ρ^{n} + 6 A ρ^{n - 1} = 0

A ρ^{n - 1} {ρ^{2} - 5 ρ + 6} = 0

\Rightarrow ρ = 3; 2 \Rightarrow a_{n} = A . 3^{n} + B . 2^{n}

n = 1 \to 3 A + 2 B = 10

n = 2 \to 9 A + 4 B = 26 {\begin{cases} A = 2 \\ B = 2 \end{cases}

a_{n} = 2 . 3^{n} + 2 . 2^{n} = 2 . {3^{n} + 2^{n}}

Answered by OlafThorendsen last updated on 22/Jul/20

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

(r - 3) (r - 2) = 0

r_{1} = 2 and r_{2} = 3

a_{n} = λ 2^{n} + μ 3^{n}

To find λ and μ

we use the initial conditions .

a_{1} = 2 λ + 3 μ = 10

a_{2} = 4 λ + 9 μ = 26

\Rightarrow λ = 2 and μ = 2

a_{n} = 2 . 2^{n} + 2 . 3^{n} = 2^{n + 1} + 2 . 3^{n}