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(20 points) An employee joined . company 2015 with . starting yearly salary of S50.000, Every year this employee receives raise of; S]OOO plus 5o0 of the salary of ...

Question

(20 points) An employee joined . company 2015 with . starting yearly salary of S50.000, Every year this employee receives raise of; S]OOO plus 5o0 of the salary of the previous year; Part I5 3 . challenge'Determine the recurrence relation for this employee $ salary Vean after 20[ and use relation compute this employee $ salary for 2016. 20[7. 201S. and 2019. Find explicir fommula for this employee'$ salary years after 2015, verify that computes his salary for 2019 correctly, and using

(20 points) An employee joined . company 2015 with . starting yearly salary of S50.000, Every year this employee receives raise of; S]OOO plus 5o0 of the salary of the previous year; Part I5 3 . challenge' Determine the recurrence relation for this employee $ salary Vean after 20[ and use relation compute this employee $ salary for 2016. 20[7. 201S. and 2019. Find explicir fommula for this employee'$ salary years after 2015, verify that computes his salary for 2019 correctly, and using this formula compute this employee s salary 2020. Write general explicit formula for any employee that has any starting salan (S) insiead of SS0.000. any base raise (b) instead of SJOOO. and any percentage increase (r) insicad of 50 0.



Answers

An employee joined a company in 2017 with a starting salary of $\$ 50,000$ . Every year this employee receives a raise of $\$ 1000$ plus 5$\%$ of the salary of the previous year.
a) Set up a recurrence relation for the salary of this employee $n$ years after 2017 .
b) What will the salary of this employee be in 2025$?$
c) What will the salary of this employee be in 2025$?$ Find an explicit formula for the salary of this employee $n$ years after 2017 .

We're told that an employee at a company starts with salary of 50,000 dollars and this promise at the end of each year. Her salary double the salary of the previous year, and there will be additional $10,000 for each year she has been with the company. In part, they were asked to find a recurrence relation for her salary for her endear of employment So a n B salary for 10th year. So first remove the salary doubles from the previous year. So this is two times and minus one, which is the salary from the previous year. Secondly, you know extra $10,000 is added breach of the N minus one years. She's been with the company so 10,000 times, and she's doing the company for end minus one years. And it follows that her salary in the anterior is going to be two times a n minus one waas 10,000 times and minus one. And we're told that being the first year, the salary is $50,000 and so we have the initial condition. A one is 50,000 in part B. Rest solved the relation to find salary for the in theory of employment notice that this is a linear, non homogeneous recurrence relation. So this has the associative linear, homogeneous recurrence relation of a n equals 2 a.m. minus one and this has characterised equation. AR minus two equals zero with characteristic route r equals two. So we have the general form for a solution to the associated homogeneous equation is Alfa Times two to the end where alphas some constant you have the non homogeneous part of the relation is F event rules 10,000 times n minus one, which is equal to 10,000 times end minus one times one to the end. And we have that 10,000 times and minus one is a polynomial of degree one and we have that one is not a characteristic route. So the general form for a particular solution is p one end plus p. Zero time is one to the end, which is the same as p one n plus p zero. You find values of the coefficients. We plug our particular solution back into the non homogeneous equation. So we have one left side p one n plus P zero On the right side. We have two times p one times n minus one plus p zero, plus 10,000 times n minus one. So V F P one n plus P zero is equal to two p. One plus 10,000 times in and we have plus negative two p one plus two p zero minus 10,000. So zero is equal to P one plus 10,000 and plus negative two P one plus p zero minus 10,000 and so we have. The P one is equal to negative 10,000 and P zero is equal to two p. One plus 10,000. This is going to be negative 20,000 plus 10,000 which is negative 10,000. And so the particular solution is negative 10,000 and minus 10,000. And so we have. The general solution to the non homogeneous equation is the general solution to the Maginness equation plus the particular solution and this is going to be Alfa Times two to the end, minus 10,000 and minus 10,000. And because their initial condition is a one equals 50,000 we have that 50,000 is equal to to Alfa minus 10,000 minus 10,000 So we have that to Alfa is equal to 70,000. That Alfa is equal to 35,000. Therefore, solution to the non homogeneous relation is 35,000 times two to the end, minus 10,000 times end minus 10,000. And this is the expression for her salary in the ampere.

This problem we're talking about a salary increase and we're going to compare two types of salary increases. So the initial value of the salary for this person is $30,000. So then for the first situation it says that every year he gets a raise of $2000. So we're gonna add 2000 every time. So to get our next term. So 30,000 plus 2000. Okay gives me 32,000. And then if I take 32,000 for the next year and add two more 1000 I get 34,000. So this would be the pattern and this sequence would be an arithmetic sequence. Because we're adding every time. So I'm gonna write this as an explicit function because they're easier to graph and understand. So mine um A. N. Is going to be 30,000 plus okay 2000 times in minus one. So this is the formula that I'm going to use to see um what happens in this scenario. Now the second scenario that were given, instead of adding 2000 every time we are told that he is given the razor, this person has given the rays of 5% a 5% raise every year. So 30,000 is your one. And then in year two we're going to multiply that by 5%. Thanks. So we got to change that to a I'm decimal point. Okay, but that's how much we're getting combined on here. We're adding 30,000 back to it. So this is 30,000 times one. Okay. We've got a common ratio here. I mean not common ratio, the greatest common factor of 30,000. So I can pull that out and combine my one and my, so if I pull it out, I've got 30,000 times one point oh five because my one plus my point oh five would get added together be 1.5. That's one way to look at it. The way that I tend to look at it instead of I'm doing that is that we're dealing with percent here. So because I'm adding to the percent, so this right here is my 100% of the salary. Next year, he's going to add 5% more to his salary, so his salary will now be 100 and five percent. And when I change that to a decimal it's 1.5. So every year he's going to be getting an increase of 5% of whatever his initial amount was. So it's going to be a compounded interest. And so my formula for a N is going to equal 30,000 times 1.5 to the power of n minus one. This is my explicit formula and this one is geometric. If he's getting a percentage raise then his geometric when he was getting an adding raise. Its arithmetic. Uh huh. So now I'm going to graph these two on the free numb works calculator. Mm Because graphing sequences is so much easier on this calculator. So here's the calculator. This is what it looks like. Here's the home screen. I'm gonna arrow down to get two sequences. Okay? And I'm gonna add a sequence. Okay. And because I've used explicit formula for both of my sequences, I can do you in. Okay. So I set up a sub in like the book uses um in programming we use use a pin or Visa been. So my first equation was 33 What was 300? I'm not 300,000. 30,000. Okay. Plus 2000 times my m minus one. So that is my um first sequence that I'm going to graph. My second sequence that I'm going to graph is going to be 30,000 times 1.5 raised to the power I used the wrong but in their back space raised to the power of in minus one. Yeah. Now I can go down and tell it to plot the graph and I have it on auto so you can see here's what they look like. If I go over and look at the table mm yep. I want I'm just typing in here because I um erase them while ago and so now I'm typing them in as I type them in notice it's generating um the next term in my sequence and showing me okay what I get. So notice that at 10 years, okay if I were if this person were to take a $2000 raise every um here at 10 years he would be making more because he was getting $2000 every year. But at year 14 it changes. So you're 14. Um This person would be making $56,000. But if I were using the geometric and doing the 5% raise every year, then he would now be making more on the geometric sequence. And you can see that it's skyrockets. I mean there's a bigger difference. So here at 20 years there's an $8000 almost $9000 difference between um just adding 2000 every year and adding a percentage race. So um which salary is higher at the beginning of the 10th year? Well that would be the first one where you're adding 2000 every year. But by the 20th year it would have been a better choice to do the percentage increase. So if you plan to stay at your job long term that's what you need to um Look for a percentage increase. If it's a short term then doing a flat rate increase is going to be better and here's how they're graft on the same one. Okay I have it on auto but I can go ahead and change if I want my why maximum to be bigger than this I can put it up 100,000. So I can see. Mhm. Um How it changes and I'm going to change my x. Maximum 2 30 years. Let's see what happens. Notice that here like I said at 14 years it's that it changes direction and the blue one now is on top of the red one on my graph. Mm. So there you go. There's um it graft on the same screen.

Heart, I suppose, to find their curse of definition of est un Remember The recursive definition of a sequence is the definition of a sequence requiring knowing the previous term in order to find the current term. So we're trying to find our recursive definition of the sequence S n. In order to even do that, we must find our very first term of the sequence. In this case, we'll call our very first term as zero s zero would be the value 30,000. Because in our problem, it tells us that the starting salary is $30,000 a year. So knowing that I recursive formula starts with us and minus one because we have to know the previous term in order to find the current term. We know that every year we have to add $2000 to our salary. So we write the recursive formula, but fine. By finding the previous term plus $2000 for this year, we have to add the previous year plus $2000. So this is a recursive formula. Now that we have our formula, we can go ahead and find our next two terms in order, find our fifth year of employment. We have to find our s one term as two s three s for in order to get her ass five term because we have your curse of definitions. So let's start by finding our US one term, that would be $30,000 plus $2000. So 32,000 dollars, we can a random. Remember this another way by saying, Okay, we have a one here. We need to simply plug in the one for this end. So s one minus one is equivalent to saying s zero plus 2000 at zero, which is $30,000 plus $2000 is $32,000. Using this method, we can keep going to find our remaining terms. Okay, so we found our terms from s one through us five. But our answer is actually what we have marked cheers us for Because, remember, it's asking us to find our salary in the fifth year of employment. So, technically, our salary for the first year of employment would be $30,000. So you're one your to your three. Your four year five, even though this is asked for This is actually our fifth year salary because technically, our zero year counts as our first year. So you solved their problem. We found part B and part, eh?

Okay, so we got somebody named I'm gonna say who? We started a new job with a salary of $400 after every year it increases by 10% for six years. So I'm gonna say you're one $400. The weekly salary thio increase by 10%. Go ahead. And right over here you need to multiply 10% times 400 and that gives us a 40 and say that is how much more money he made the following year. And so 40 plus 400 is 440. So first year, second year, let's go ahead and find four more by again, almost by 10% times 440 this time. All right, woman, time for you guys. 10% times 440 is 44. And then I'm gonna go ahead and add that to 440 to get 484. Okay, I'm not gonna write it this time, but I can multiply 0.10 times 484 which gives me $48.4 40 cents. It's all right, that number. And now I need to add that to 484 and we see that 484 Fleiss 48.4. Think I might have hit multiple on my calculator. 532 dollars and 80 cents, Which can't be true, because that's a 40 cents on the double check. 38.4 plus for 84 0 yeah, I got a right, except it's 40 cents. 90 cents. Excuse me. Uh, see 12342 more years. So let's take this multiplied by 0.1. And his job isn't gonna gonna increase by $53 in 24 cents. So I'll be very careful. As I say, that is what we need to add to this number here. So $532.40 leads us to $585 in 64 cents One more time and this was probably gonna have to have some rounding. Um, if I multiply 10.10 times this, I get that his job is gonna increase in the weekly salary by $58.56 rounding to the nearest cent. Feel free to check on your calculator. If you ever want to check my numbers. Um, but I'm gonna add that $585 in 64 cents and get that in that six year, this salary will be $644 and 20 cents. And the last thing we have to do say, write a recursive definition, and I'm gonna go ahead and say, right now you're one is gonna be $400 for a weekly salary. But your end, all you have to dio is maybe I should say n plus one. I think that's the best way to do these things, n plus one. Or, in other words, the next year you take the previous year and you multiply it by. How about a right? Were you most bye bye First 0.10 for 10% and you add that two that previous year solid. So there's your a cursive definition


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