Understanding Characteristic Roots in Recurrence Relations

Welcome to my channel. In today's video we will discuss method of characteristic root to solve linear recurrence relation with constant coefficient. Before understanding this method, you should know what is recurrence relation and what is linear recurrence relation. I have made a video based on this and I will provide the link of playlist in the description. I discussed these terms in the video 1 of this playlist. So you listen to it before listening to this video. If you are new to the channel, then subscribe to the channel and press the bell icon so that you can get the notification of the upcoming videos first. If you like the video, then like it, comment and tell me, I get motivation from all of them. And if you like it, then share it so that others also like it and help them. So let's start. So we have to learn this method of characteristic rules for linear recurrence relation with constant coefficient. So I told you in that video that linear recurrence relation is like this. Its form is like this. that a n a n minus 1 to a n minus these terms will be there in this in which in multiplication what happens constant c naught c1 these are all constant and equal to fn any function of n ok so now it is divided into two categories how homogeneous recurrence relation second is called non homogeneous recurrence relation so who is called if this function fn is 0 then it is called homogeneous recurrence relation and if it is non zero then it is called non homogeneous recurrence relation so there are two categories and But today we will talk about homogeneous recurrence relation. First I will tell you its steps and following these steps we will learn to solve some questions. And then lastly I will give you unsolved questions which you will solve, you will practice and you will gain confidence. So let's start. See, what is the first step? We put an equal to r power n, an minus 1 equal to r power n minus 1 and so on. We put it like this. and I am telling you with surety that you will do 1 or 2 questions and you will find it very easy so understand carefully so first we write the equation in R means we put the terms only so we get an equation from it so I have written the second step find an equation in terms of R this is called as characteristic equation and auxiliary equation this equation is called as characteristic equation or auxiliary equation then what we do in the next step we solve this equation so I have written solve characteristic equation and find characteristic roots means characteristic equation which will be solved and the values will come, they are called characteristic roots then if characteristic roots, now see this is good to explain to you, understand this carefully, I am going to tell you a very useful thing that we have taken out the roots, suppose the roots are R1, R2, R3, all three are different, they are distinct, so how do we write the general solution An? B1 R1 power n, B2 R2 power n, B3 R3 power n, B1 B2 B3 is constant What we did is, we wrote the power of the roots of the root NNN and multiplied one constant by the other in the second term and the third in the third term. Okay. Suppose if there are two, then we write only this much. If there are three, then we write three. If there are four, then we write four. Okay. But suppose if we repeat R1 R1, then what will we do? Now R1 R1 is twice, so we will not write twice that R1 has power N, R1 has power N. We will not write like this. What will we do? But if there are two, then two constants should also come. So we will write B1 plus NB2 and. R1 power n If you remember, when we solve first order differential equation When we read CFPI, something like this happens C1 plus XC2 So here also it is the same B1 plus n B2 R1 power n Write it twice, not only once But there will be two terms in the bracket Now you may also think that if it was coming three times, then what would we do? Assume R1 R1 R1 If it was coming three times, then what would we do? We write three terms, how? B1 plus n B2 plus n square b3 and r1 power n because it is repeating so it will come only once but 3 constants will come how will 3 constants come? in this n will be multiplied for second n came, for third n square if there was one more then n cube would come then if there was one more then n power 4 would come ok, we will write like this so what happens is our an because to solve the recurrence relation we have to find an ok, now see what I have taken as the next possible deal if suppose 2 r1 would be there 1 r2 would be there 1 r3 would be there means 2 are same and the rest 2 are different so what happens? So, we wrote the same as I told you above, we wrote R2 separately, I wrote it wrong, R2 R3, so here R2 will come, here R3 will come, okay, so the power of R2 is N and AB3 is constant, then the power of R3 is N and B4, okay. Now, it may also happen that someone will write R2 for R3 first and then write for it later, so the one who will write for R2 first will write B1 after that, the one who will write for R3 will write B2 after that, then the rest will write B3 and B4, so it doesn't matter, you can write in any order. It doesn't make any difference. Next, I have given 3 R1 and 1 R2. So, 3 R1, as I told you earlier, b1 plus n b2 plus n square b3 will be r1 power n and then we will write r2 separately for that b4 because it has come to 3 so we will write a new constant it is not like you have written 3 here so you have to write 3 here you will get the same constant as the different roots so 3 is this so we wrote b4 for the fourth and wrote r2 power n so now you must have understood how to write root sum an now see you will enjoy next question we have taken here is I have taken an equal to this and we have to solve it. With initial condition. So first we will solve it and then we will see the initial condition. So we have written this equation and we have taken both the terms there. Now what we told you, what we told you in the first step, I told you to put rn instead of an. So now you are doing this for the first time, so I will tell you in 2-3 steps. But next time when you will do it, you will write this equation directly. How did you see? First we wrote an instead of an, this came. a n minus 1 here n minus 1 is done so minus r power n minus 1 is done this is minus 2 and instead of this r power n minus 2 is equal to 0 ok now what to do if we take rn common so what is left 1 in this r power minus 1 means 1 upon r in this what will be left minus 2 and 1 upon r square is equal to 0 this is come now we take LCM so this term will come inside and the term which will come in upon will go to 0 so this is come ok now this will not be 0 if it will be 0 then the solution will be 0 we have to find non-trivial solution So we have to keep the terms of this bracket as 0 and find the value of r. So we have to find the value of r. So what is the value of r? 2 and minus 1. So I just told you about the distinct value. The roots that come, the value of r, as we were doing here. If the value is distinct, then what was done? Write constant and the power of this root is n. Then the second constant and the power of the second root is n. So we wrote constant and what is the root? 2. So the power of 2 is n plus constant is b2. and root is minus 1 so if we don't have initial condition then our work is finished we will do further for initial condition but after solving recurrence relation our solution came before moving forward I want to say something that the 1,2,3 steps that I have written there is no need to write it we can write it directly listen carefully see how much is its order its order is going till 2 n-3 so its order is 2 If it is more than 3, then it becomes 3. Because we have to see the difference between highest and lowest. I told you in the previous video. So, it is going to 2, n-2. So, this quadratic equation will be made in it. Of order 2. So, how will we do it? In the shortcut, what will we do? Instead of an-2, we took only a constant term. Minus 2. Take minus 2. It has 1, so 1r will come. If there is no constant, then it is not. Minus came. here r square will come means in the last one only constant then r then r square it will keep increasing like this or it can go in reverse if it is going till 2 order 2 then first r square term then r term then constant so we can write it directly so if you write this equation directly practice it then you will save time and you can solve it with the help of calculator or you can do it manually so we know the solution of recurrence relation now it has given us initial condition so we will use it also condition is a0 is equal to 0 and a1 is equal to 1 so what does a0 equal to 0 mean? when n is 0 then the value is 0 so we put n as 0 here, when n is 0 here then a0 is equal to 0 so a0 is equal to 0 here so we write 0 here so what is the power of 2? 1 so what will come? b1, so what is the power of minus 1? 1 so what will come? b2, so 0 equal to b1 plus b2 Now we have to put 1 in place of n. The subscript indicates that you have to take n. So it indicates that put n as 1. So put n as 1 in the solution. So when we put n as 1, it becomes a1 equal to b1 2 plus b2 minus 1 power 1. And a1 is given as 1. So 1 is on the left and 2 is b1 and this is minus b2. I will tell you again after writing what I did in this. I put n as 0 in this, then what happened? a0 equal to b1. 2 power 0 is 1 only, plus b2. If minus 1 power 0 is 1, then this happened. And a gave us 0 because I put 0 in place of it. So in this way, two initial conditions were given, two equations came. We will solve them, we will find b1, b2. And we found b1, b2, so the value of b1, b2 came here and we put it back in it. So we put the value of constant here. So we put 1 by 3 instead of a1 and b1. 2 power n as it is. Then we put the value of b2. And power n as it is. So this is the solution of recurrence relation. So initial condition is given. So it will come like this. If it is not given then b1, b2, b3 will come in constant terms. Okay. Let's see the next example. Next example we have given. a1 equal to 4 a1 minus 1 minus a1 minus 2. Initial condition is also given. So first we took these two terms there, so this is done. Now what we do directly, we put an equal to r power n. So you can see directly, this is the degree of 2, so here we start from r square, so instead of this, r square minus 4, 1r, that is, reduce the power, plus 4 equal to 0. Order 2 was the same, I started from r square, this equation came. Okay, let's match, see this is written. So you don't need to write anything, you can write directly. Now just say this much, we put an equal to this, putting this, you are getting this. So you can write this directly. If we solve this, then 2 and 2 roots are repeated. So I told you that if we repeat, then how do we write? So what will we take? We will take 2 constants. We will take b1 like this. b1 plus nb2 and 2 power n. If these are 3, then what do we write? b1 plus nb2 plus n square b3. Right? We write that then we write 2 power n. So now we have written this. Then the initial condition is a0 equal to a1 equal to 1. So what does a0 equal to 1 mean? When we put n as 0. so we will put 0 here, then a will be 0, a is the value of 0, so 1 is here, then if we put 0 in this, then b1 and this 0 will be 0, then b1 will be left empty, this came and if we put 0 in this, then this 1 is done, so only b1 is found from here to keep n as 0, so b1's value is 1 straight. Then next time we have given a1 equal to 1, so instead of n, we have to put 1, so if we put 1, then a1's value is 1, so 1 came to the left, what will happen on the right side, if we put 1, then b1, 1 b2 means b2 and 2 power 1 means 2. So this equation came, we know the value of b1, so we find the value of b2. So now we put both values of b1 and b2 back in the equation of an and we got to know this. So the solution came an equal to this 1 minus n by 2 dhu ki power n. So this is the solution of this recurrence relation. Let's see one more example, I have taken two more. Now see there is no condition given in this. It is given to you to solve it. So to solve it, I have written directly. putting an equal to r power n now see order is 3 so start from r cube so r cube came then minus 8 as it is then plus 21 then minus 18 so direct equation came if you solve it then 2,3,3 will come you can do it with calc and there are more shortcuts to solve you can do it with that too 1 is different and 2 is repeated so this is different 2 so for this we have written b1 2 power n So, as we have done in the previous lesson, b2 plus nb3 will be written. Now, b1 is used, so we will not write b1 again. So, we will write b2 plus nb3. Now, we take a new constant and which is the root? 3. So, 3 power n. So, constant always comes and the root power is n. Okay, keep in mind that it is repeated or distinct. So, it is done. We have not taken the condition, so it is done. Now, see the next one is very good. Order is of 3 and condition is also given. So, now see. We have taken all the terms on one side, now see how we have written the equation. See here the order is given 3, so it means we have to start with cube, so what happened with R cube, then subtract it, plus R square, then minus, then subtract one more, then 4R and minus 4 equal to 0. So we wrote this equation directly. Now we will solve it. So I have done this by taking the common here, you can find the roots directly from the galaxy, so here came minus 1, 2 and minus 2. If all three are different, then we have written. write this again b1 of b1 is power of minus 1 and then b2 is power of 2 and then b3 is power of minus 2 now we will do the same thing we were doing for initial condition earlier now see three initial condition a0 equal to 8 a1 equal to 6 a2 equal to 26 so first initial condition means n has to be put zero so when zero is put then a is zero means 8 so 8 came in left and what will come in right put zero so this b1 this one is b2 because this term will be 1 on n equal to zero then this will also be term 1 If it becomes 1 on 0, then it is B3. One equation is here. Now take n equal to 1, so put 1 on both sides, so a1 is 6, so 6 is left, if we put 1 here, then what will happen, if we put 1 here, then it will be minus 1, then minus b1. If we put 1 here, then it will be 2, so 2b2. If we put 1 here, then it will be minus 2, so minus 2b3. Okay, so a2 equals 26, so we will put 2 on both sides, so this is 26, it came to the left, now put 2 here, so minus 1 power 2 is 1, so it means b1 came. Then put 2 here. So, if 2 is equal to 4, then 4 is b2. If we put here, then 4 is b3. We solved all three of them. We found the value of b1, b2, and b3. So, when we found all three of them, then we put them back in this. We put the solution we had taken out in this. So, our solution came. Okay. So, this is very, very easy. I hope you understood this. These two questions are written by me. You can take a screenshot of them. Answers are also given. and solve them and tell me by commenting whether you can solve or not whether you know how to find a solution or not so do tell me by commenting how you liked the video because I have discussed it on homogeneous and I will make it on non-homogeneous and there are many types of FNK so I will cover all the cases of it I think there are 3 or 4 cases I will cover that so tell me by commenting how you liked the video if you liked the video then do like it and subscribe to the channel if you haven't subscribed yet and share it with your friends. Take care, Radhe Radhe.

I have made a video based on this and I will provide the link of playlist in the description. I discussed these terms in the video 1 of this playlist. So you listen to it before listening to this video.

If you are new to the channel, then subscribe to the channel and press the bell icon so that you can get the notification of the upcoming videos first. If you like the video, then like it, comment and tell me, I get motivation from all of them. And if you like it, then share it so that others also like it and help them.

So let's start. So we have to learn this method of characteristic rules for linear recurrence relation with constant coefficient. So I told you in that video that linear recurrence relation is like this. Its form is like this. that a n a n minus 1 to a n minus these terms will be there in this in which in multiplication what happens constant c naught c1 these are all constant and equal to fn any function of n ok so now it is divided into two categories how homogeneous recurrence relation second is called non homogeneous recurrence relation so who is called if this function fn is 0 then it is called homogeneous recurrence relation and if it is non zero then it is called non homogeneous recurrence relation so there are two categories and But today we will talk about homogeneous recurrence relation.

First I will tell you its steps and following these steps we will learn to solve some questions. And then lastly I will give you unsolved questions which you will solve, you will practice and you will gain confidence. So let's start. See, what is the first step?

We put an equal to r power n, an minus 1 equal to r power n minus 1 and so on. We put it like this. and I am telling you with surety that you will do 1 or 2 questions and you will find it very easy so understand carefully so first we write the equation in R means we put the terms only so we get an equation from it so I have written the second step find an equation in terms of R this is called as characteristic equation and auxiliary equation this equation is called as characteristic equation or auxiliary equation then what we do in the next step we solve this equation so I have written solve characteristic equation and find characteristic roots means characteristic equation which will be solved and the values will come, they are called characteristic roots then if characteristic roots, now see this is good to explain to you, understand this carefully, I am going to tell you a very useful thing that we have taken out the roots, suppose the roots are R1, R2, R3, all three are different, they are distinct, so how do we write the general solution An? B1 R1 power n, B2 R2 power n, B3 R3 power n, B1 B2 B3 is constant What we did is, we wrote the power of the roots of the root NNN and multiplied one constant by the other in the second term and the third in the third term.

Okay. Suppose if there are two, then we write only this much. If there are three, then we write three.

If there are four, then we write four. Okay. But suppose if we repeat R1 R1, then what will we do? Now R1 R1 is twice, so we will not write twice that R1 has power N, R1 has power N.

We will not write like this. What will we do? But if there are two, then two constants should also come.

So we will write B1 plus NB2 and. R1 power n If you remember, when we solve first order differential equation When we read CFPI, something like this happens C1 plus XC2 So here also it is the same B1 plus n B2 R1 power n Write it twice, not only once But there will be two terms in the bracket Now you may also think that if it was coming three times, then what would we do? Assume R1 R1 R1 If it was coming three times, then what would we do? We write three terms, how?

B1 plus n B2 plus n square b3 and r1 power n because it is repeating so it will come only once but 3 constants will come how will 3 constants come? in this n will be multiplied for second n came, for third n square if there was one more then n cube would come then if there was one more then n power 4 would come ok, we will write like this so what happens is our an because to solve the recurrence relation we have to find an ok, now see what I have taken as the next possible deal if suppose 2 r1 would be there 1 r2 would be there 1 r3 would be there means 2 are same and the rest 2 are different so what happens? So, we wrote the same as I told you above, we wrote R2 separately, I wrote it wrong, R2 R3, so here R2 will come, here R3 will come, okay, so the power of R2 is N and AB3 is constant, then the power of R3 is N and B4, okay.

Now, it may also happen that someone will write R2 for R3 first and then write for it later, so the one who will write for R2 first will write B1 after that, the one who will write for R3 will write B2 after that, then the rest will write B3 and B4, so it doesn't matter, you can write in any order. It doesn't make any difference. Next, I have given 3 R1 and 1 R2. So, 3 R1, as I told you earlier, b1 plus n b2 plus n square b3 will be r1 power n and then we will write r2 separately for that b4 because it has come to 3 so we will write a new constant it is not like you have written 3 here so you have to write 3 here you will get the same constant as the different roots so 3 is this so we wrote b4 for the fourth and wrote r2 power n so now you must have understood how to write root sum an now see you will enjoy next question we have taken here is I have taken an equal to this and we have to solve it.

With initial condition. So first we will solve it and then we will see the initial condition. So we have written this equation and we have taken both the terms there.

Now what we told you, what we told you in the first step, I told you to put rn instead of an. So now you are doing this for the first time, so I will tell you in 2-3 steps. But next time when you will do it, you will write this equation directly.

How did you see? First we wrote an instead of an, this came. a n minus 1 here n minus 1 is done so minus r power n minus 1 is done this is minus 2 and instead of this r power n minus 2 is equal to 0 ok now what to do if we take rn common so what is left 1 in this r power minus 1 means 1 upon r in this what will be left minus 2 and 1 upon r square is equal to 0 this is come now we take LCM so this term will come inside and the term which will come in upon will go to 0 so this is come ok now this will not be 0 if it will be 0 then the solution will be 0 we have to find non-trivial solution So we have to keep the terms of this bracket as 0 and find the value of r.

So we have to find the value of r. So what is the value of r? 2 and minus 1. So I just told you about the distinct value. The roots that come, the value of r, as we were doing here.

If the value is distinct, then what was done? Write constant and the power of this root is n. Then the second constant and the power of the second root is n.

So we wrote constant and what is the root? 2. So the power of 2 is n plus constant is b2. and root is minus 1 so if we don't have initial condition then our work is finished we will do further for initial condition but after solving recurrence relation our solution came before moving forward I want to say something that the 1,2,3 steps that I have written there is no need to write it we can write it directly listen carefully see how much is its order its order is going till 2 n-3 so its order is 2 If it is more than 3, then it becomes 3. Because we have to see the difference between highest and lowest. I told you in the previous video.

So, it is going to 2, n-2. So, this quadratic equation will be made in it. Of order 2. So, how will we do it?

In the shortcut, what will we do? Instead of an-2, we took only a constant term. Minus 2. Take minus 2. It has 1, so 1r will come. If there is no constant, then it is not. Minus came.

here r square will come means in the last one only constant then r then r square it will keep increasing like this or it can go in reverse if it is going till 2 order 2 then first r square term then r term then constant so we can write it directly so if you write this equation directly practice it then you will save time and you can solve it with the help of calculator or you can do it manually so we know the solution of recurrence relation now it has given us initial condition so we will use it also condition is a0 is equal to 0 and a1 is equal to 1 so what does a0 equal to 0 mean? when n is 0 then the value is 0 so we put n as 0 here, when n is 0 here then a0 is equal to 0 so a0 is equal to 0 here so we write 0 here so what is the power of 2? 1 so what will come?

b1, so what is the power of minus 1? 1 so what will come? b2, so 0 equal to b1 plus b2 Now we have to put 1 in place of n.

The subscript indicates that you have to take n. So it indicates that put n as 1. So put n as 1 in the solution. So when we put n as 1, it becomes a1 equal to b1 2 plus b2 minus 1 power 1. And a1 is given as 1. So 1 is on the left and 2 is b1 and this is minus b2.

I will tell you again after writing what I did in this. I put n as 0 in this, then what happened? a0 equal to b1.

2 power 0 is 1 only, plus b2. If minus 1 power 0 is 1, then this happened. And a gave us 0 because I put 0 in place of it.

So in this way, two initial conditions were given, two equations came. We will solve them, we will find b1, b2. And we found b1, b2, so the value of b1, b2 came here and we put it back in it. So we put the value of constant here.

So we put 1 by 3 instead of a1 and b1. 2 power n as it is. Then we put the value of b2. And power n as it is.

So this is the solution of recurrence relation. So initial condition is given. So it will come like this. If it is not given then b1, b2, b3 will come in constant terms. Okay.

Let's see the next example. Next example we have given. a1 equal to 4 a1 minus 1 minus a1 minus 2. Initial condition is also given.

So first we took these two terms there, so this is done. Now what we do directly, we put an equal to r power n. So you can see directly, this is the degree of 2, so here we start from r square, so instead of this, r square minus 4, 1r, that is, reduce the power, plus 4 equal to 0. Order 2 was the same, I started from r square, this equation came.

Okay, let's match, see this is written. So you don't need to write anything, you can write directly. Now just say this much, we put an equal to this, putting this, you are getting this.

So you can write this directly. If we solve this, then 2 and 2 roots are repeated. So I told you that if we repeat, then how do we write? So what will we take? We will take 2 constants.

We will take b1 like this. b1 plus nb2 and 2 power n. If these are 3, then what do we write? b1 plus nb2 plus n square b3.

Right? We write that then we write 2 power n. So now we have written this.

Then the initial condition is a0 equal to a1 equal to 1. So what does a0 equal to 1 mean? When we put n as 0. so we will put 0 here, then a will be 0, a is the value of 0, so 1 is here, then if we put 0 in this, then b1 and this 0 will be 0, then b1 will be left empty, this came and if we put 0 in this, then this 1 is done, so only b1 is found from here to keep n as 0, so b1's value is 1 straight. Then next time we have given a1 equal to 1, so instead of n, we have to put 1, so if we put 1, then a1's value is 1, so 1 came to the left, what will happen on the right side, if we put 1, then b1, 1 b2 means b2 and 2 power 1 means 2. So this equation came, we know the value of b1, so we find the value of b2.

So now we put both values of b1 and b2 back in the equation of an and we got to know this. So the solution came an equal to this 1 minus n by 2 dhu ki power n. So this is the solution of this recurrence relation. Let's see one more example, I have taken two more. Now see there is no condition given in this.

It is given to you to solve it. So to solve it, I have written directly. putting an equal to r power n now see order is 3 so start from r cube so r cube came then minus 8 as it is then plus 21 then minus 18 so direct equation came if you solve it then 2,3,3 will come you can do it with calc and there are more shortcuts to solve you can do it with that too 1 is different and 2 is repeated so this is different 2 so for this we have written b1 2 power n So, as we have done in the previous lesson, b2 plus nb3 will be written.

Now, b1 is used, so we will not write b1 again. So, we will write b2 plus nb3. Now, we take a new constant and which is the root?

3. So, 3 power n. So, constant always comes and the root power is n. Okay, keep in mind that it is repeated or distinct. So, it is done.

We have not taken the condition, so it is done. Now, see the next one is very good. Order is of 3 and condition is also given.

So, now see. We have taken all the terms on one side, now see how we have written the equation. See here the order is given 3, so it means we have to start with cube, so what happened with R cube, then subtract it, plus R square, then minus, then subtract one more, then 4R and minus 4 equal to 0. So we wrote this equation directly.

Now we will solve it. So I have done this by taking the common here, you can find the roots directly from the galaxy, so here came minus 1, 2 and minus 2. If all three are different, then we have written. write this again b1 of b1 is power of minus 1 and then b2 is power of 2 and then b3 is power of minus 2 now we will do the same thing we were doing for initial condition earlier now see three initial condition a0 equal to 8 a1 equal to 6 a2 equal to 26 so first initial condition means n has to be put zero so when zero is put then a is zero means 8 so 8 came in left and what will come in right put zero so this b1 this one is b2 because this term will be 1 on n equal to zero then this will also be term 1 If it becomes 1 on 0, then it is B3. One equation is here. Now take n equal to 1, so put 1 on both sides, so a1 is 6, so 6 is left, if we put 1 here, then what will happen, if we put 1 here, then it will be minus 1, then minus b1.

If we put 1 here, then it will be 2, so 2b2. If we put 1 here, then it will be minus 2, so minus 2b3. Okay, so a2 equals 26, so we will put 2 on both sides, so this is 26, it came to the left, now put 2 here, so minus 1 power 2 is 1, so it means b1 came.

Then put 2 here. So, if 2 is equal to 4, then 4 is b2. If we put here, then 4 is b3.

We solved all three of them. We found the value of b1, b2, and b3. So, when we found all three of them, then we put them back in this.

We put the solution we had taken out in this. So, our solution came. Okay. So, this is very, very easy.

I hope you understood this. These two questions are written by me. You can take a screenshot of them.

Answers are also given. and solve them and tell me by commenting whether you can solve or not whether you know how to find a solution or not so do tell me by commenting how you liked the video because I have discussed it on homogeneous and I will make it on non-homogeneous and there are many types of FNK so I will cover all the cases of it I think there are 3 or 4 cases I will cover that so tell me by commenting how you liked the video if you liked the video then do like it and subscribe to the channel if you haven't subscribed yet and share it with your friends. Take care, Radhe Radhe.

Transcript for:Understanding Characteristic Roots in Recurrence Relations

Transcript for:
Understanding Characteristic Roots in Recurrence Relations