Answer to Question #117133 in Combinatorics | Number Theory for Priya

Question

Find the solution of the recurrence relation a_n=2a_(n-1)+〖3.2〗^n.

Answer 1

Characteristic polynomial: x - 2 = 0 roots is 2 , 〖3.2〗= 3 =>

a_n = 2ⁿ*a₀ + 3ⁿ + 2*3^n-1 + 2²*3^n-2 + ... + 2^n-1*3 =>

a_n = 2ⁿ * (a₀ + 1.5 ⁿ + 1.5 ^n-1 + ... + 1.5) =>

a_n = 2ⁿ * (a₀ + 1.5 * (1.5 ⁿ - 1) / (1.5 - 1)) =>

a_n = 2ⁿ * (a₀ + 3 * 1.5 ⁿ - 3)