Suppose that a sequence satisfies the following J0 0 J1 1

Suppose that a sequence satisfies the following: J_0 = 0, J_1 = 1, and J_n+2 = J_n+1+ 2J_n for n = 0, 1, 2, ... Solve the difference equation and apply the initial conditions to find an explicit formula for J_n

Solution

J_n+2 = J_n+1 + 2J_n

n= 0 , 1, 2, 3....

J₀ =0 ; J₁ = 1

let us take n=0 , then J _{2 =} J₁ + 2J₀

  = 1 + 2(0)

J₂ = 1

n= 1 , J₃ = J₂ + 2 J₁

   = 1 + 2 = 3

n= 2 , J4 = J3 + 2 J2

= 3 + 2(1) = 5

so the series is 0, 1, 1, 3 ,5, 11.......