If U_ n+1 =3 U_n-2 U_ n-1 and U₀=2, U₁=3 , then U_n is equal to

Step-by-step solution

The given recurrence relation is U_ n+1 - 3U_n + 2U_ n-1 = 0 . The characteristic equation is r^2 - 3r + 2 = 0 . Factoring the quadratic equation, we get (r-1)(r-2) = 0 , which gives roots r₁ = 1 and r₂ = 2 . The general solution is of the form U_n = A(1)^n + B(2)^n = A + B 2^n . Using the initial conditions: For n=0 , U₀ = A + B = 2 . For n=1 , U₁ = A + 2B = 3 . Subtracting the first equation from the second: (A + 2B) - (A + B) = 3 - 2 B = 1 . Substituting B=1 into A+B=2 , we get A = 1 . Therefore, U_n = 1 + 1 2^n