00:01
All right, so for this one, we are given that a .n is equal to negative 3 times a.
00:11
N minus 1, minus 3, a.
00:18
N minus 2, and then minus a minus 3.
00:24
And so n would need to be greater than are equal to 3 in order for this to make any sense.
00:29
And so we'll start with the roots characteristic equation.
00:33
So we'll let a .n be equal to r cubed.
00:38
And so a.
00:39
N minus 1 is going to be r squared.
00:44
A .n.
00:44
Minus 2 will be r.
00:47
R.
00:47
And then the a .n.
00:51
Minus 3 will just be equal to 1.
00:54
And so if we substitute these values into the equation given, we end up with r cubed minus is equal to negative 3r squared minus 3r minus 1.
01:12
And when we move everything over to the left hand side, we get r cubed plus 3r squared plus 3r plus 1 is equal to 0.
01:26
And then we factor, and when we factor this, we get r plus 1 plus 1.
01:31
Cubed is equal to 0.
01:35
So this means that our roots are negative 1, but with multiplicity 3, right? so three times.
01:50
Okay, so we'll use those roots to build a solution to our recurrence relation, and that solution will be of the form, a .n is equal to alpha 1 times negative 1 to the n, plus alpha 2 times negative 1 to the n plus alpha 3.
02:17
And this should be due to the multiplicity, this will actually be times n.
02:24
Okay, then alpha 3 times negative 1 to the n, but then multiplied by n squared.
02:34
Okay, so we'll use our initial conditions to solve for this.
02:37
So we know that a not is equal to 5...