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[Music] okay welcome to fan formula for the sequence 0 1 0 1 0 1 0 1 and so on forever as we can see this is just a repeating sequence with 0 & 1 0 & 1 and so on right ok before to anything that's happened pension first right here this is our first term and I will have this to be an is equal to 1 alright and the next one is when n is equal to 2 and then n is equal to 3 and then so on forever of course we could have done it with n is equal to 0 to begin with but let's just use n is equal to 1 to start with okay okay here's the idea whenever we're trying to grab formula for any sequence that's repeating most of the time we can just use sine cosine because that's no periodic functions so it's going to produce repeating sequence by in this case we can avoid using that as well and that's do that first ok so here we have 0 1 0 1 0 1 and so on let's look at this part first if we take negative 1 and put it in a parenthesis raised to the nth power we are not talking about functions so we can actually have a negative number for the base if you're talking about sequence because the end I just passed the poll numbers right for our choices of the n values so this is ok if this is what we have let's see when n is 1 we will have negative 1 to the first power now will give us negative 1 for the first term right and when n is 2 we get parenthesis negative 1 squared that will give us positive 1 and then likewise for the rest so as you can see this will give us negative 1 1 negative 1 1 negative 1 1 and so on right and look at even though the first term is negative 1 but the next hour-- when n is equal to 2 we have 2 1 which is similar to that right and then the best part is this right here it does repeat just like the original the one that we want to find out okay so we are making a good progress but I really want all the art terms right here to be zeros rather than negative one okay we have negative one how can we make negative one equal to zero we know negative one plus one is zero so you'll be so nice if I can just add one to this and then we'll be having a zero right now let's look at another part of the sequence so this is just going to be the constant sequence namely just one alright as I said this is a constant sequence and if you would like you can put on one and then raised to the nth power and then you see what n is 1 you get 1 to the first power is 1 well then it's true you get 1/2 a second Pope which is still 1 and then so on forever right and of course 1 to the nth power is always 1 so you can just erase this part this is just a constant sequence and now if we add both sides here is the formula part right let's put it down as one go first and then we add it with this negative 1 to the nth power like that looking at this we will see it's going to produce negative 1 plus 1 0 and then 1 plus 1 is 2 and then 0 again to again 0 again to again and this right here keeps on going forever right and you see how cool this is because this is almost the same as that but instead of the 2 we want to have the 1 right well 2 divided by 2 is 1 and the reason I want to say divide in stuff sub top one is because I don't want to mess up the zeros so now we can just look at the defense side and divide this by 2 which you will imply will justify all the terms by 2 isn't it and now we will see that's the formula at the end we will have this 1 plus negative 1 to the nth power over 2 this formula will produce 0 and then 1 and then 0 and then 1 and then and one and this will keep on going forever this is good for n is equal to one two three four five and so on right so here is an answer for the formula for this sequence and now let's take a look of how we can also use sine cosine to help us out okay here's another way to approach the sequence and keep in mind whenever we are dealing with a repeating sequence most of the time we can use a trig function for it the one that we have is 0 1 0 1 0 1 and so on right this is kind of like running around circle isn't it because if repeating and you see when you are running around circle that should remind you of the unit circle right so now let's just take a look of the unit circle right here okay so of course you know you're just running one circle that's why use a trig function because this is circular this is periodic right it repeats and you know here are four important points right this point here is 1 comma 0 this point right here is 0 comma 1 and this run right here is negative 1 comma 0 and this one right here is 0 comma negative 1 and it keeps repeating right okay and this is of course 0 radians and this right here is of course PI over 2 radians and this is PI and then 3 PI over 2 and then of course you can also say this is the same as 2 pi and so on forever of course you can also go this way but partly directions this way ok so now we are kind of running on one circle how would you like to run on one circle okay this right here we start with 0 if you look at this point you want to use this point first then you can talk about the 0 here which is the sine value right so perhaps we can do that this right here it's a sine value okay so let's see I just want to stop with time with you guys alright well I know that starting from this point I go to the next one it will be one right if you're looking at the white value namely the site value zero one zero that's good for the first three terms and then the next one will be negative one but I want to have past the one how can I make a negative 1 to be a positive one it's okay just take the absolute value right so that's good and see every single time when I jump from here to here I had to go up by PI over 2 right and the PI over 2 PI over 2 and so on so we can just take the sine function and each every time we go up by PI over 2 so it's a multiple of PI over 2 times whatever that you are talking about now here's the technical part this right here is when n is equal to 1 though right so be sure you want to have n is equal to 1 if I do plug in n is equal to 1 here in fact you are talking about sine of PI over 2 you're actually looking at this point here but I want to start right here so that's easier 0 1 0 and so long right well this is PI over 2 I can just shift it back right here so I can do it look at this and -1 to that how is that aha once again I want to start with 0 even the whole the index right here start with 1 but that's ok I can just do a minus 1 and what n is equal to 1 this is 0 and you're talking about this point right and then when n is equal to 2 PI over 2 times 2 minus 1 which is 1 so you are talking about PI over 2 and the sine for you right here is 1 and then the next one will be 0 and then negative 1 and so on right you don't want to have any negative numbers because of this so you can just do the absolute value yeah like that right this it's also going to produce 0 1 0 1 so that's it for the sine function this is the answer for this sequence why not use cosines wheel let's take a look suppose if I won't use cosine well this sequence start with 0 1 0 1 in fact we'll be looking at this here because this right here it's 0 if I start with 0 here the next won't be negative 1 it's ok I can take absolute value and then 0 and then 1 and then 0 and then negative 1 and then 0 and then 1 and so on right ok each every time I still go up by PI over 2 so I will just put up PI over 2 in this case in fact I just need to have an N this is it and don't forget to take the absolute value as well this right here when n is equal to 1 plug into here you get cosine PI over 2 which you are talking about this point here right and that's going to give you 0 and when n is equal to 2 you're talking about cosine of PI which is right here it's negative 1 but the absolute value will make everybody happy so you get oh this right okay anyway listen that are the answers as well and you can also try it with tangent and now here's the challenge for you guys find a formula for the sequence 0 0 1 0 0 1 0 0 1 and be sure you come when you answer down below alright and I will have another video for you guys as up now that's it good [Music]