Question

19. Solve the following recurrence equations using the characteristic equation. (a) T(n) = 2T(5/+10g3 n T (1) =0 for n > 1, n a power of 3 (b) T(n) = 10T()+12 T (1) =0 for n > 1, n a power of 5 or nI, na power of 5 (c) nT (n) (n 1)T(n-1)+3 for n> 1 T(1) = 1 (d) nT(n) = 3 (n-1)T(n-1) _ 2 (n-2) T (n-2) +4" T (0) 0 T (1) =0 for n > 1

##Solve for D only

Answer 1

n T(n) = 3 (n - 1) T (n - 1) - 2 (n - 2) T (n - 2) + 4^x Let F (n) = n T (n), then the equation reduces to F (n) = 3 F (n - 1) - 2 F (n - 2) + 4^n Homogeneous pant F (n) = 3 F (n - 1) - 2 F (n - 2) characteristic equation x^2 = 3x - 2 Rightarrow x^2 - 3x + 2 = 0 P x 4^n - 1 - 16/3 4^n - 2 + 4^n Rightarrow (x - 2) (x - 1) = 0 4^n - 2 [2 - 1/3 + 1] Rightarrow x = 1, 2 Therefore solution of homogenous pant = A 1^n + B 2^2 = A + 2^n B \r\n now Let c 4^n be a particular solution of f (x) Rightarrow c 4^n = 3c 4^n - 1 - 2c 4^n - 2 + 4^n dividing by 4^n - 2 we get c 4^2 = 3 c 4 - 2c + 4^2 Rightarrow 16 c = 12 c - 2c + 16 Rightarrow 6c = 16 Rightarrow c = 8/3 Therefore F(n) = 8/3 4^n is a particular solution