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Acme Annuilies recently offered an annuily that pays 8.1% compounded monthly: What equal monthly deposit should be made Into this annuity In order to have 5159,000 ...

Question

Acme Annuilies recently offered an annuily that pays 8.1% compounded monthly: What equal monthly deposit should be made Into this annuity In order to have 5159,000 In years?The amount of each deposit should be (Round to the nearest cent )

Acme Annuilies recently offered an annuily that pays 8.1% compounded monthly: What equal monthly deposit should be made Into this annuity In order to have 5159,000 In years? The amount of each deposit should be (Round to the nearest cent )



Answers

Annuity If a deposit of $\$ 100$ is made on the first day of each month into an account that pays $6 \%$ interest per year compounded monthly, determine the amount in the account after 18 years. Annuity Refer to Exercise $77 .$ Show that if the monthly deposit is $P$ dollars and the rate is $r \%$ per year compounded monthly, then the amount $A$ in the account after $n$ months is given by $$ A=P\left(\frac{12}{r}+1\right)\left[\left(1+\frac{r}{12}\right)^{n}-1\right] $$

In the given question it is being said that a person is going to pay $100 each month. Okay. Each month our director told two and the 80's but in them and up to five years and the compounding is done monthly. Okay compounding is very monthly. So you know the formula for final amount should be, it was too Principal multiply one bloods are Brian to the power anti now let his depositing the first installment and you can say that the first installment will provide us the first installment will provide us Interest for 16 months. So the changes into 100 multiplied one plus The rate of change that is 0.02 Upon 12 minutes 12 multiplied by 60. Okay now the second installment we can say that it will provide you interest Only for 59 months since the month is going to be reduced. So so one happens and finally last year my we can say that will provide us Interest for only one month. OK now just we have to simply heard all these amounts so to formula to wear these amounts. Okay, we can say their tier first amount was 100 multiply one plus zero point later to about 2020 power 60 2nd amounted to 100 multiply one plus 0.0 to divide. Well to it. to the bar 59. So on going up last one was 100 multiply one plus 0.2 About where to the power from here. You can say that it is. You're 12.02 divided by 12 24 60. This is handled multiply 12.021 1224 59. So on. Going on and multiply one plus 0.02 to the vault. Well, we're too late. This can be witness 12 100 or two by 12 Department. Now we'll do somehow 60 times here. I assume this is the first time. And assume this other last time. Okay, so while summing up anthems, we know that formula is a multiplied one minus R to the power and upon one minus R. So here we have to sum up 60 times. That's formula has to be here. First of all, we have to take the first time. I can say that in the first time is 100 multiply. Well for mosquito to develop very well Into 1- are combined ratio. Your common ratio is 12.02 led by 12. Multiplied by N. Okay, so your n is 60 upon one minus R means one minus Well from 0 to divide by. Well. Okay. Or you can associate it. This formula can be no rudeness After multiplying by a 100 It gets you. 12 point sorry, 12 0 to upon 12 and into 12.02 upon 12 to the power 16 minus one. A bone. 1200 to about 12 -1. Okay, so I'm using now our calculators. And you can see that here. It comes out to be zero point to a 0.0.12. I am going on the third page. Our value was 12 year or two. The world. World. Well into where the Well, it was To help the middle to upon 12 with about 60. 12.02 By the world with about 60 -1. And in a bone We get 0.02 x 12. Okay, you can look here regularly. No, after canceling this too. Well, I am using the small scientific calculator. I could be loud. 12 0 to buy 20-2. Okay, I have done that. And in brackets I had in brackets I have to do £12 or 12. 12 to devour 60. Okay. 12.0 to divide by 60. Wait a while. Well one go to related to bring 60 to be power including our 60. And yeah, we have to rule my next one. Okay. I can say that here. I have to do minus one and break it close. So I had done that. Well, they don't do 12 0 to divide by 102 and 12.02 divided by 12. Here I have to do 12. Okay. We were pretty about 16 minutes. So the value that comes out of 63 15.24 means it is 63.63 15 63 15.243 Approximately Approximately these here. 1243 1243 in dollars. So this is my final amount that I should get after five years. Okay, thank you.

So this question says that you're making a deposit at $100 at the beginning of each month and into an account that pays 3% interest compounded monthly. So it tells me that my formula is 100 onto one plus 0.3 divided by 12 plus dot, dot, dot and then plus 100 onto one plus 0.3 divided by 12. Race the 60th power. So if we recognize that this is actually can be written as the sum for men equals 1 to 61 of 100 onto one plus 0.0 3/12. Um, sorry. 1 to 60. Race to the end. Then we'll be able to find our formula, because this is just a geometric sequence, okay? And what I'm gonna do for the sake of just doing the calculation purge is I'm gonna ask Dez most to do that calculation. So I'm going to go to from one to 60 okay? And you may not have access to ah calculator like this, and they have to go through the calculation a little bit differently, but I'm just gonna do it this way, and then I'll show you what you might be able to do as well. Um And then my answer is there. So it's 64 80 83. The other option is to recognize that that is can be written as 100 onto one minus R R. Is that so? We can go 0.3 divided by 12 and then add one and we get one minus 1.25 raise to the 60th and then divided by one minus 1.25 Okay, and that will give us the some. Although you may take this 60 and make it a 61. Actually, because of the formula, the geometric sequence formula, we want this to be n minus one. Right? So you have to add in a neck stra 61 then subtract 100 bucks because you're not actually contributing $100. Um, that last time you're just getting interest on it. So if you do that, you will get the same answer. You get 65 80 30 to minus 100 then you get 64 80 32. So you get the same answer using that formula as well

So for this question, we want to find the balance when the interest is compounded both monthly and continuously. So we'll start out with monthly from the given information. We know that PC 1 75 you know there are R is equal to 0.55 I just converted the percent to a decimal, and we know that our time is 25 years. And so our formula for our account balance when interest is compounded monthly is p times 12 over our times. The quantity one plus are over 12, the 12 t minus one. So once we plug in our known values, we get 75 times 12/0 750.55 times the quantity one plus zero 0.5 5/12 to the 3/100 power minus one. We should get that. Our balance is 48,152 0.80 for part B. We want to make sure the interest is compounded continuously. That formulas a equals p times the quantity Eat the R T minus one all over e the are over 12 minus one. We can substitute in our given values. We have 75 times e to the 1.375 power minus one all over 0.459 We should get that. Our account balance for part beat is 48,245 0.7

So for this question, I've already written out the formulas that will be using for Parts A and B. So let's start with part a part. A. We want to figure out the account balance where the interest is compounded monthly. And so we are given that P is equal to 30 r is equal to 0.6 or 6% and rt is 50 years. And so we're gonna go ahead and plug in our known values into our formula. We have a equals 30 times 12/0 300.6 times the quantity one plus 0.0 6/12 to the 600 power minus one. If we simplify that we get A is equal to 113,615 0.73 for part B. We're going to be doing the same thing. Except our interest is compounded continuously so we can substitute in our known values, just like we did for part A. We have 30 times e to the 0.6 times 50 if power minus one that quantity all over e to the 0.6 divided by 12 minus one If we simplify that, we get that A is equal to 114,000 227.177


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