Solve the recurrence relation an 5an1 6an2 where r a0 0 a

Solve the recurrence relation: a_n = 5_an-1 - 6a_n-2 where r a_0 = 0, a_1 = 6

Solution

a_n = 5a_n-1 - 6a_n-2     and a₀=0, a₁=6

therefore a_n-5a_n-11+6a_n-2 =0;

writing charactristic equation

x²-5x+6=0

(x-3)(x-2) = 0

x=3 or x=2;

a_n=A3ⁿ + B2ⁿ   where A and B are constant

at n=0;

0=A+B ----> A= - B;

at n=1

6= 3A + 2B = 3A-2A = A     therefore A=6 B = -6

a_n=63ⁿ -62ⁿ = 6( 3ⁿ - 2ⁿ )