Question

Solve a0 = 1; a1 = 6 and an = 6an-1 - 9an-2 + n; for n greater than or equal to 2?
Find Recurrence Formula....

Answer 1

You have the recurrence formula. You need a[n] in terms of n. Right

Answer 2

This is is a difference equation problem

Answer 3

The homogeneous equation is
\[
a_n = 6 a_{n-1} - 9 a_{n-2}
\]

Answer 4

The characteristic equation is
\[p(t)=t^2-6 t+9=(t-3)^2
\]
It has a double root r=3.
The solution of the homogeneous equation is
\[
a[n_]= ( A + B\ n) 3^n
\]

Answer 5

I mean
\[
a_n= ( A + B \ n )3^n

\]

Answer 6

Now you have to find a particular solution of the non homogeneous equation of the form F n + G.
You solve and you find
\[
F=\frac 1 4 \\
G=\frac 3 4\\
\]

Answer 7

The general solution is
\[
a_n= ( A + B \ n )3^n +\frac n 4 + \frac 3 4

\]

Answer 8

To find A and B, use the initial conditions.

Answer 9

You find
\[
A= \frac 14 \\
B= \frac {17}{12}

\]

Answer 10

The final answer is
\[

a_n=3^n \left(\frac{17
n}{12}+\frac{1}{4}\right)+\frac{n}{4}+\frac{3}{4}
\]

Answer 11

Thank you so so much! This helps more than you could know. The way you described things really makes sense to me too. You're a lifesaver :)