Challenging Precalc!! Please help me! Long assignment Pg. 1: Sequences and series can be used to approximate or model financial investments. We can use them to see how much an investment will increase or decrease over a certain amount of time. At first, we can discuss some banking language that is used to describe how investments change over time. And investments interest "compounds "every time money is added or taken out of an account. A percentage refers to the amount of increase or decrease of the investment. And the compounding period explains how often the interest is compounded in the investment. So saying A certain investment "Compounds monthly at a rate of 10% "means that 10% of the current value of the investment is added to the investment total every month. Let's move on to creating equations to describe this increase. Let's give a new example and let you try to be creative mathematically to come up with your own way of finding the total amount of money at the end of a certain time period. -Example: given that $100 investment compounds annually (once a year) and has a positive interest rate of 4%, discuss a method or write an equation that would find the value of the investment after one year. Give the answer as well if you can. > Pg.2: And so now we can see that if we just add these values up, we will get the total interest earned from the investment after however many years we'd like to specify. And even add that total to the initial investment, to figure out the total amount and investment. So now knowing we have a sequences and series application here, if we wanted to find A specific amount of interest added it to the investment in a particular year after the initial investment was made we can use the following equation: (capital i) I n=ir(1+r)^(n-1) Where "I n" is the interest obtained only in year 'n', 'r' is the interest rate as a decimal, 'I' is the initial investment, and when multiplied by 'r' is the interest gained after the first year. This is the same equation you will see in this chapter except we have the (1 + r), since our common ratio is in terms of the interest rate, which must be added to 1 first to make sure we are talking about an increase sequence, not a decreasing one. We also have i*r as the initial value in our sequence. Now if we wanted to find the total amount of interest gained in a certain number of years we can use this equation: S n=-i(1-(1+r)^n) Where S n(superscript) is the sum of 10 years of interest and all the other variables are the same as the above equation. Again, we see a very similar equation to what you are used in this chapter, only a few differences as we still have to have the (1 + r) as our common ratio. And because our initial term in the sequence is i*r, The denominator cancels out, and we are left with the above equation. Pg.3: OK so now let's expand this into the real world using examples from real world investments. Millions of stocks are bought and sold each day through many different stock exchanges. Stocks can be bought through a stock exchange, for a certain cost per stock. As the price of the stock increases by a certain percentage, once you sell that stock, you acquire that percent of your original investment. Stocks can also go in the opposite direction, and you can also lose money based on how much the price of stock decreases as a percent. In this case we are only going to look at the positive side of the situation where the annual returns are positive percentages, not negative. The specific stock SPX:IND, which is an abbreviation for the S&P 500 stock is like a collection of 500 companies stocks combined. It is one of the most popular stocks to buy or sell, in fact you might see it scrolling on the bottom of the screen during a broadcast on television. As of when this project was written, the one year return was at +24.67%, which is quite high. The one-year return refers to the percentage of the stocks increase or decrease based on the price of the stock calculated exactly one year ago. So if someone was to invest $1000.00 into that stock exactly one year ago, then a 24.67% year to date return means that your money would have increased by $246.70. Alright, now let's use this example and test the equations we have above. Pg. 4: let's say we wanted to find the amount of interest you would get in only the fourth year if the annual interest was a consistent 24.67% and we initially put in 1000.00 $ into the investment. Which equation do we use? And once you know that, what is the interest? Great work. Now let's find the total amount of the investment after the four years. Remember first we need to find the total interest gained by using the sum equation above then add that to the initial investment. Do that work here > Great work! OK, in this next section we're going to use the skills you've learned and you're going to find some stocks yourself. Stock 1 -company name: -stock abbreviation: -One year return: Stock 2 -company name: -stock abbreviation -one year return Pg.5: OK, now we are going to use the equations above to figure out what the total amount we'd have including the initial investment if we put in $5000 into each of these stocks for five years. Take The one year return as a consistent interest rate. Use the space here to figure it out for stock 1: > Stock 1 total: Now do the same for stock 2: > Stock 2 total: Pg.6: Great, and now we are going to figure out if it is more important to find a better interest rate, or invest for a longer period of time. So first let's double the interest rates (or one-year returns) of each stock. Record the new one year returns here: > Stock 1 doubled one year return: Stock 2 doubled one year return: And so let's do the same thing we did earlier. So we have $5000.00 put into each stack, find the total amount in the investment after five years using the new, doubled, one year returns as the interest rate. Use the space here to find the total of the first stock: > Stock 1 total with doubled interest rate: And now for the second stock: Stock 2 total with doubled interest rate: And now you're going to do the same thing except we will double the time. Instead. So using the original one year returns you recorded earlier, an initial investment of $5000.00 in each stock, and time period of 10 years, find the total amount in the investment. Do stock one first: Stock 1 total with a double the time period: And now here's space to do the same for stock 2: Stock 2 total with a double the time period: Pg.7: Now we want to compare the values we found here: > Stock 1 Stock 1 total with doubled interest rate: Stock 1 total with double the time period: Stock 2 total with doubled interest rate: Stock 2 total with double the time period: For stock one, which value is larger, the one with the doubled interest rate, or the doubled investment.? By how much? And for stock 2 again, which value is larger? And by how much? Reflection Knowing the results of the last section of this assignment what would you say is more important, a longer investing period, or a larger interest rate? Let's move away from the example of stock investments, and simply discuss saving money in a savings account. Again, do you think it's more important to put more money into a low interest savings account a week/month/year, or continue putting money into the account for a longer period of time