On Friday during class we were in the cafeteria because the math room was having work done in it. In the cafeteria we worked with our peers around us to help us get set for Monday. There was 2 problems. The first problem was

You decide to begin contributing to a savings/retirement fund. You find an annuity with projected annual earnings of 6% compunded quarterly. (In other words, 1.5% every 3 months.) You will make a quarterly payments of $600.

All we had to do was figure out and write down the pattern. The pattern my group came up with is as follows.

A1=600

A2= 600+(1.015(600))

A3=600+(1.015(600)+(1.015(600)squared)

A4=600+(1.015(600)+(1.015(600)squared)+(1.015(600)cubed)

The pattern works and we will discuss in detail what goes on during Mondays class.

The second question is very similar to the first problem. The second problem is below

Suppose you take 500 mg of acetaminophen (Tylenol) and continue taking 500 mg every 4 hours. About 59% if the drug is eliminated by your body every 4 hours (so 41% remains). Assume that each dose immediately enters your system. Below is the pattern we found.

A1=500

A2=500+(.41(500))

A3=500+(.41(500))+(.41(500)squared)

A4=500+(.41(500))+(.41(500)squared)+(.41(500)cubed)

On monday we will discuss what these patterns mean and how this is relating to what we will be learning the next few days.

Today in class we just went over the logarithms test so I’ll just bring up a few things about sequences.

1. A sequence is an ordered list of numbers

2. For our purposes they are either arithmetic(adding the same value), geometric(multiplying by the same value), or neither

3. They can be written as recursive or explicit functions(As addressed in Emily’s post)

4. Remember that when your doing problems that the first term might be t(1) or t(0) so it’s important to pay attention to that

5. If you want to find the nth term just plug n into either your explicit or recursive methods

6. If you want to find out what term a given value is set it equal to your equation(solving will require, everyone’s favorite, logarithms for geometric functions)

If you understand these basic things about sequences you should be fine. On the bright side our class can’t do any worse then we did on the logarithms test. Good luck with series tommorrow everyone.

Scotty P.

(Just a side note .9 repetant is equal to 1)

If a sequence is neither arithmetic or geometric then a matrix can be used to write an equation for it.  I will admit, I barely remember how to use a matrix considering we learned them freshman year.  The good news is that they aren’t hard to pick up on again!

Let’s use the following example because it is neither arithmetic or geometric.  The sequence goes like this:

4, 10, 18, 28, 40, 54, 70…

Notice that the differences between each of these numbers aren’t constant. 6, 8, 10, 12, 14, 16… But, the differences of the differences is constant, 2.  This means that the equation for the sequence has a squared value.   We can use the quadratic formula, which is tn=an²+ bn+c (where n is the term number and tn is the value in the sequence).  Since there are three unknowns (a, b, and c), we need to three equations.

4=a(1²)+b(1)+c
10=a(2²)+b(2)+c
18=a(3²)+b(3)+c

Now, this is where the matrix come in.  We use the general equation Ax=b (where A is the coefficient, x is the variable, and b is the solution).

Notice how the first row corresponds with the first quadratic equation, the second row with the second equation, and the third row with the third equation.  Now, in order to get the x column all by it self we would have to divide b by A, which is the same as multiplying b by the inverse of A.  This can be done by putting the matrix in your calculator.  First, go to second inverse, which is the matrix key.  Scroll over to edit.  Make sure A is 3×3, then fill in the numbers into the matrix.  Now edit b, make sure this is 3×1, then fill in the numbers.  Go to the home screen.  To get the solution, hit the matrix key then select A.  Now, hit the inverse key to get the inverse of A.  Go back to the matrix menu and select b.  Hit enter, and you should get 1, 3, and 0.  You plug these numbers in for a, b, and c in the quadratic equation.  

tn=1n²+ 3n+0 (n is the term number)

We now have the equation for the sequence!

Today in class we had to claim problems from section 14.2. The claimable problems were numbers 2,4,9,13,14, and 20. Since no one claimed 20 and because I wasn’t paying attention when 9 was presented, I will go over 2,4,13 and 14. My apologies to Jake for not paying attention to his presentation but I’m sure it was great. To save you the time of scrolling down past all of this just to see who the next scribe is, it will be Scoot (yes I spelled it that way on purpose)

For problems 2 and 4 you had to

a. tell whether the sequence is arithmetic, geometric, or neither

b. write the next two terms

c. find t100

d. Find the term number of the term after the first elipsis marks

2.  27, 31, 35, . . . , 783, . . .

a. because the sequence increases by a consant (4), it is arithmetic

b. to find the next two terms just keep adding four, (39, 43) 

c. you can create an explicit formula for the sequence, (tn=27+4*n) and just plug 100 in for n to find the 100th term, 427

d. use the explicit formula to find n when 783=27+4n. n=189

4. 100, 90, 81, . . . , 3.0903… , . . .

a. because each term is formed by multiplying the pervious term by a constant (.9), it is geometric

b. multiply the previous term by .9, (72.9, 65.61)

c. the recursive formula is tn=100*.9^n, (.00265…)

d. 3.0903=100*.9^n, you solve for n and you get the equation .030903=.9^n, to solve this you need to use your knowledge of logarithims YA! You know that log(.030903)=n*log(.9), solve for n and n=33

13. An office building is worth $1,300,000 and it’s value depreciates every year.

a. The building depriciates by $32,500 every year. Find the first few terms. What kind of sequence is it? How much would it be worth after 30 years? When will it be worth nothing?

Just subtract 32,500 from 1,300,000 three times (1296750, 1235000, 1170000). It decreases by an constant so it’s arithmetic. The explicit formula is tn=1300000-32500*n, so when n=30, tn=325000. 0=1300000-32500*n, n=40.

b. The building depreciates at 10% a year. Find the first few terms. How much will the building’s value decrease the first, second, and third year? When will the value decrease by less than $32,500?

Creating a chart of years, value, and how much the value decreases will help you solve this problem. The first few terms are (1170000, 1053000, 947700). The values it decreases by are (130000, 117000,105300).  Because the amount the building’s value decreases by 10% of the previous term, the equation to find when the value decreases by less than 32500 would be 325000=1300000*.1*.9^n. using logarithms, you solve for n and get that n=15.

14. You decide to save money by putting $5 into a piggy bank the first week, $7 the second week, $9 the third week, and so forth.

a. What kind of sequence is this? How much will you deposit on the 10th week? When will you deposit $99?

It increases by a constant so it’s arithmetic. The explicit formula is tn=3+2n so when n=10, tn=33. 99=3+2n, n=43.

b. What is the total amount you saved at the end of the 10th week? Show how you can find this by averaging the first and tenth values and multiplying this average by the total number of weeks.

 5+7+9+11+13+15+17+19+21+23=140. ((5+23)/2)*10=140

c. How much would you have saved in a year?

There are 53 weeks in a year so you would save $2912

So that was today’s class. I hope this helped and if your looking for the next scribe, you obviouly missed it in the first paragraph. Good job being observant!