Mumbai University > Computer Engineering > Sem 3 > Discrete Structures

Marks: 7 Marks

Year: May 2014

The characteristic equation of the recurrence relation is $r^3+3r^2+3r+1=0$ Its roots are r=-1. Hence the sequence {$a_n$} is a solution to the recurrence relation if and only if

$a_n=α_1 (-1)^n+-α_2 (n)(-1)^n+α_3 (n \times n)(-1)^n=α_1+α_1n+α_3(n \times n)$

For some constant $α_1$ and $α_2$.

From the initial condition, it follows that

$a_0=5=α_1 + α_2(0) + α_3(0) \\ a_1= -9 = - α_1 - α_2 - α_3 \\ a_2= 15 = α_1 +2α_2 + 4 α_3$

Solving the equations, we get $α_1=5, α_2=3, α_3=1$

Hence the solution is the sequence {$a_n$} with

$a_n =5 + 3n + n^2$