[Music] hello class in today's video we're going to be going over modeling compound interest using recursive relations the learning intention of this video is we're going to have a introductory in the understanding of what in geometric sequences are and lastly we're going to be able to represent a model compound interest problems using recursive relations similar to simple interest there are two ways that we could actually express the balance of an investment that earns compound interest so similarly to a simple interest you can do this using recursive relations which is remember the long way method recursive meaning it's repeated so you're going to have to add on or repeat certain steps to get your final answer and lastly you could also do this using the explicit formula and this was done in the last lesson so in the last lesson I would have shown you the compound interest formula and you were going through several examples of how you could actually apply that particular formula the main intentions of this video is we're just going to be looking at how do we actually represent this using the recursive relations but before I do that I do want to quickly go through again what is the differences between simple interest versus compound interest so if you look at the diagram below you notice that for every single year you're going to be earning the exact same amount of interest so you're always going to be earning a flat rate or 50 dollars per year whereas compound if compound interest is a bit different because the amount of interest it gets reinvested and that interest also earns interest on top which is why the amount of interest per year always increases as you can see over here now if you look on the right hand side I said since the amount of interest earned in each successive year for compound interest is different or it's not constant this means that it doesn't actually follow the similar pattern which you would generally see in an arithmetic was okay so if I also write the terms or the final amount of each year this is what it'll look like we know that our simple interest follows like an algorithmic sequence kind of pattern because it has the same common difference so in this example over here the common difference is 50 because you're always going to be earning 50 so I can kind of guess what the other values are in the sequence or as for compound interest this doesn't follow the same trend so there is no static common difference at all and because of that this actually follows something referred to as a geometric sequence okay and in January metric sequence is characterized by where the next term increases by the exact same right or the exact same percentage in this case I'm here this is referred to as a geometric and you're going to learn more about this in the followings especially in year 12 now as you can see if here because it doesn't actually follow the arithmetic ones pattern at all we actually have a different general formula to actually represent compound interest in this case so all compound interest problems can be represented by the following recursion equation up here the main difference is that instead of actually adding it or actually multiplying this instead okay so please copy this down inside your exercise book and again the main thing that you need to note is that this R over here is the interest rate but that is always going to be expressed as a percentage now if I'm going to show you side-by-side the main differences between the recursive equation for simple interest versus compound interest as I said earlier on the main difference is that for a simple interest you're gonna be adding the common difference all day every single time or as for compound interest what you're doing instead is you're gonna be multiplying so this is just the main differences between these two recursive equations over here so let's look at an example so you have a better understanding so enough first example today it says the following recursive relation can be used to model a compound interest investment of $1,000 paying interest at the rate of 8 percent per annum okay and they're very giving you the recursive relations in the recursive relation VN represents the value of the investments after n years okay partyers reads use the recursive relation to find the value of the investment after one two and three years now before I've been answering this question I'm just going to quickly go how do they actually obtain this recursive relation equation so this case of easier remember it represents your principle so that's this amount over here and the Ray that they obtained 1.08 because you need to know how they got this formula is that that remember you have to by your villain by our and to calculate our that was one plus the interest rate over 100 in this case the interest rate was eight so if you go one plus eight over 100 this gives you 1.0 H and that's and that's the reason why they've kind of x over y n alright so now let's just finally answer Part A so for part eight if you want to calculate the amount in after the first year you need to know the the initial amount so therefore it would be V one is equal to one point eight times V zero whatever v-0 is equal to so these are in this cases you can want to one thousand dollars so if you put this into a calculator you're going to get 1080 dollars do the same thing for B two so B 2 if you want to find a V so you don't have to multiply the 1.08 by V 1 so if you do this one point nine eight multiplied by one thousand eighty dollars is gonna equal to one thousand one hundred and sixty six dollars and forty cents and do the same thing to obtain V three that's a type of here that should be a V 3 and therefore that should be your answer now Part B reads when will the investment first exceed one thousand five hundred dollars in value so in order to do this you're pretty much gonna have to repeat this process until you're gonna see a value that's larger than 1500 so in this case have you I've already done that so in this case notice that if you keep on going you'll notice that after the sixth year in this case the amount is going to be larger than 1500 so the answer is after 60s in this case what I like either do now is I like to have an attempt to answering this question over here so if you please follow the exact same example from beforehand you should be able to get these answer credits so give yourself five minutes or so and then compare your answer with my answer alright guys if you guys done this successfully okay this is what you should have got so again V 1 is equal to 1.0 6 3 multiplied by V 0 put this check calculator you should get an answer of twenty one thousand and two hundred and sixty dollars do the same thing for B 2 as well as B 3 now to get the answer for Part B Part B reads determine how many years it takes for the value of the loan to first exceed $30,000 so again just keep keep on repeating this process and if you did this successfully you should get an answer off after seven years and if you were to kind of like prove this if you are doing this exact same step v7 in this case would be 1 plus 6 3 x v6 and you found out what v6 is then B sticks is gonna equal to twenty eight thousand eight hundred fifty five dollars and fifty seven cents and if you multiply this notice how this value now exceeds thirty thousand dollars which is why seven years is the answer and lastly for Part C plus C wants you to write down a recursive relation so write down an equation something similar to this over here to model when you have an initial loan of eighteen thousand dollars so in this case your v-0 is going to equal to eighteen thousand we therefore need to find out what the value of R is so in this case the interest rate is 9 point 4 1 plus nine point four divided by 100 is going to equal to one point eight four five and therefore I'm sorry that's a typo it should be equal to 1.0 nine four and therefore this should be equal to VN plus 1 is equal to 1.0 9 4 multiplied by V in we're going to go through another example because this example is slightly different to compare to the first example particularly for Part B and I really want to show you this because many questions in vc often confuse you at Part B so the question reads Brian borrows $5,000 from a bank he'll pay interest at the rate of 4.5% per annum let VN be the value of the loan after n compounding periods write down a recursive relation to model the value of Brian's loan if interest is compounded yearly as well as quarterly now I'm just gonna just highlight this over here cuz that's what I'm trying to do I'm trying to find out the equation in this case and notice that I've also included a general equation just for us to refer to so that we could actually answer this question okay now again we're trying to write down the relation so in this cases we I need to state what the value of v-0 is gonna be in this case B 0 is gonna equal to 5,000 so v-0 is equal to 5,000 and I've also stated what my V end represents in this case because you have to actually do this when you have different compounding frequencies I've got to mention that earlier on so now what I'll do is I'm gonna find out what the value of R is gonna be so R in this case is going to be 4 5/100 so therefore my are gonna equal to 100 for five substitute us in so therefore this over here is going to be your final answer when you rain down the recursive relations now how is this different for Part B now because you can being compounded quarterly your VN now represents the value of the loan after n quarters or so and we know that in a year there are four quarters okay so again your visa is gonna be the exactly the same but your are your interest rate is gonna be a bit different usually as you can serve here this represents the interest rate for the whole year which is why says per annum you're doing it per quarter okay and because it's per quarter you actually have to divide this by 4 because this if it multiplied by 4 point 5 that gives you the interest rate for the entire year but since you're trying to find out the value after in quarters you need a divided by 4 since there's four quarters okay so as you can see here I put this in brackets and I've actually divided that by 4 and as you can see here my interest rate is gonna be a bit different my R is now gonna be 1.0 1 1 to 5 so therefore this is gonna be my final answer so if you do in turn to a question that's in assessing monthly okay so when it's monthly it's gonna be a bit different it's gonna therefore be dividing it by 12 since there are 12 months within a year by n nearly wouldn't be the same thing since this biannually is 2 times definitely / - okay okay class what I like you to do now is I want you to have an attempt at this question over here so if you guys just please follow the example that I've shown you just beforehand you should be able to get this question correctly but again the only thing I want you to be wary of is that in this case of it is being compounded quarterly okay guys and when you do answer Part B where it says if Jessica pasted back everything she owes the bank after one year how much money will show you need to pay so just consider are you actually trying to find out what a V 1 is V 2 V 3 or what is it that corresponds to one year so you guys are five minutes and then comparing your answer with my answer okay so again if you're drinking piña you're trying to write the recurrence relationship so in this case over here you need to state what your principles gonna be and then you just need to find what the interest rate is going to be okay but again what you need to do because you tried it is compounded quarterly you do need to actually divide this by 4 since there are 4 quarters within the year itself so you can do this correctly you should get 1.0 2 and therefore just multiply and vien by 1.0 - now guys for Part B because you're trying to find that after 1 you are technically what you're trying to find is value of v4 okay so that's the value people will tell you how much money that jessica has to owed back after one year since since in this case V n represents the value after the first quarter so that means after 3 months so therefore find out what v1 is equal to v1 in this case is going to equal to three thousand five hundred and seventy okay and then we need to do the same thing to find out the v2 which gives you the balance after half a year so do this that will give you three thousand six hundred forty one dollars and forty cents do the same thing for the v3 as well as before and therefore you should get this over here as your final answer guys all right guys what I like you to do now is I only want you to answer these two questions for today's lesson and if you guys did this successfully then this is the answer that you guys should be getting for these two questions okay this is the end of todays video hopefully this video helped you out I'll see you guys again in the next video bye