Solve the recurrence relation for the variables indicated be

Solve the recurrence relation for the variables indicated below:

Solve the recurrence relation for the variables indicated below: b_n = 8b_n-1 - 15b_n_2 b_1 = 2 b_2 = 16 If the explicit formula for the sequence is written in the form b_n = u(s_1)^n + v(S_2)^n then find S_1 and S_2. u and v.

Solution

The Recurrence relation is given by

b_n = 8b_n-1 - 15b_n-2 with b₁ = 2 and b₂ = 16.

Since it is linear homogeneous recurrence relation ,so first find its characteristic equation.

r² - 8r + 15 = 0

( r - 3 )( r - 5 ) = 0

r = 3 , 5

So, the general solution of the given recurrence relation is

b_n = u(3)ⁿ + v(5)ⁿ ------------ ( 1 )

Now use the initial conditions b₁ = 2 and b₂ = 16.

From ( 1 ) , b₁ = u(3)¹ + v(5)¹

                               2 = 3u + 5v ---------- ( 2 )

and b₂ = u(3)² + v(5)²

          16 = 9u+ 25v-------    ( 3 )

Solving ( 2 ) and ( 3 ) , we get u= - 1 and v = 1

Therefore, the required solution is

b_n = - (3)ⁿ + (5)ⁿ