For each recurrence relation and initial conditions, find: (i) general solution; (ii) unique solution with the given initial conditions: (a) an = 3an−1 + 10an−2; a0 = 5, a1 = 11

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a_{n} = 3 a_{n - 1} + 10 a_{n - 2}

a_0=5

a_1=11

(i)

Characteristic equation: $r^2-3r-10=0$

(r+2)(r-5)=0

Characteristic roots: $r_1=-2, r_2=5$

The general solution is

a_n=\alpha_1(-2)^n+\alpha_2(5)^n

for some constants $\alpha_1$ and $\alpha_2.$

ii) Find the unique solution with the given initial conditions

a_0=\alpha_1+\alpha_2=5

a_1=\alpha_1(-2)+\alpha_2(5)=11

\alpha_1+\alpha_2=5

7\alpha_2=21

\alpha_1=2, \alpha_2=3

The unique solution with the given initial conditions is

a_n=2(-2)^n+3(5)^n