00:01
So for this question, it's pre -calculus, right? we're looking at recurrence relation and characteristic equations.
00:11
So let's go in and start by writing what they give us out, right? so they give us a sub n is going to equal to negative 10 times a sub n minus 1.
00:22
And then it's going to be minus 21 times a sub n minus 2.
00:31
And then we're also given that a sub 0 is equal to 2 and a sub 1 is equal to 1.
00:39
So now that we're given these things here, the first thing we can do is go ahead and just simply by looking at what we're given.
00:49
We can realize that we have to use a characteristic equation for this problem here, right? because a sub 0, what's given is a sub 0 and a sub 1, and each of these are looking at n minus 1 and n minus 2.
01:04
So those are the continuations, right? so once we realize that, we can go ahead and use the characteristic equation, g of x equals a sub 0 plus a sub 1 times x, and then plus our sum to infinity of n equals to a.
01:24
N equals a .m.
01:25
Equals x to the power of n.
01:27
And that will be our characteristic equation.
01:33
So now we can substitute in all these things that we're given, right? and we'll get g of x equals 2 plus a sub 1 is 1, so plus x plus the sum, n equals to infinity of, oh, right, so n equals to infinity of, you'll get negative 10, a sub n minus 1, minus 21, a, and a sub n minus 2.
02:05
And then you can basically split this part up into two different summations, and by doing that, you'll get here minus 10, right, you can take that out, and they can say n equals two, infinity times a sub n minus 1, and then the split times, of course, x to the power of n, and then you'll have that and then times x to the power of n here minus 21 times the summation of n equals 2 because we're taking the 21 out here and the 10 out here to infinity of a sub n minus 2 times x to the power of n.
02:45
Right.
02:47
From there, now that we have it in this much simplified form, we can actually realize that, sorry, we can actually realize that what this is going to be is we can actually take, or sorry, i think i missed a step here.
03:11
Oh, yeah, when you plug, when you, after you plug things in, this will actually become x to n minus 1 and x to n minus 2 because those are terms for n that you're starting.
03:21
And so here you'll start with x to the n where n equals 1 and then here you'll start with actually know your server 2 here as well so when then when you plug this in right then you'll get 2 plus x minus 10 times x here right 2 when you plug 2 in for x you'll get uh just normal x and then you plug 2 in here you'll get uh a sub um a sub 1 and and and that can be represented by this characteristic equation minus a sub zero, right? and this is just part of the algebra here.
04:00
And then you do the same thing over here.
04:03
And you'll end up with over here x squared times g to the x.
04:13
And then from there, you do some more algebra, and you can get it to 2 plus x minus 10x times g of x minus 10.
04:25
X a sub 0 um and a a a sub zero um and a a a sub zero is uh just two so they'll get 20x here right uh 10 minus 10 x g of x plus 20 x um the minus minus we'll make you a plus and then minus 21 x squared times g of x so now that we have all that um this is going to be of course still g of x is equal to all this right so then g of x if you combine all the g of x terms to one side, you will get 1 plus from there, 10x, right, and then plus 21x squared from this side.
05:18
So now we have all that, and then we leave these things on this side, so you have 2 plus 21x...