### 14.2 Arithmetic, Geometric, and other Sequences

Hey Guys! I hope everyone is coping sorta okay with AP's coming up and everything. In 5 weeks it will be summer!

Ok so lets define some terms first:

**Sequence**is a function where the domain is a set of positive integers. The independent variable is*n*and the dependent is tn.**A Recursion formula**specifies tn as a function of the preceding term (tn-1)**A Explicit formula**of a sequence specifies tn as a function of*n*.

**Arithemetic Sequence** is where you add a constant to the previous term to get the next term. The constant added is called the** common difference**. (This can be adding a negative number)

**Geometric Squence** is where you multiply the previous term by a constant (**the common ratio**) to get to the next term. (you can multiply by a fraction which is basically dividing)

**Formulas:**

It is good to use recursive just to find the next few terms but for finding terms later in the sequence use the explicit formula. Also, a sequence can have a finite or an infinite number of terms depending on whether its domain is finite or infinite.

**Arithmetic Linear Function:**

- Explicit: Tn = To + D(
*n*-1) - Recursive: Tn = Tn-1 + D

so To is the initial value in the sequence, R is the common difference, Tn is a value in the sequence, and Tn-1 is the previous term.

**Geometric Exponential Fution:**

- Explicit: Tn = To * R^(
*n*-1) - Recursive: Tn = Tn-1*R

so To is the initial value in the sequence, R is the common ratio, Tn is a value in the sequence, and Tn-1 is the previous term.

**Calculator!**

- Make sure you are in sequential mode.
- go to y=
*n*Min=1 (you will usually want to leave this =1)*u*(*n*)= this is where you put your equation (u(n) is the same as t(n))- to get
*u*go to 2nd 7 and for the*n*press the regular x button. - for
*u(n*Min)= put the first term of the sequence - once you plug in your equation you can go to table set to find values for any term. Go to table set to start at a specific value.
- you can also graph additional sequences under (
*v*) and (*w*) (the same way) in case you want to compare the sequences.

**Examples!**

Given the following problem how would you plug it into the calculator (i had a hard time with this so i am giving it as a example problem.)

You put $1000 into the bank with 6% interest a year. What are the steps for putting this scenario into the calculator.

- seq mode
- y=
- then under nMin=1 (leave this alone)
*u(n)=*u*(n-*1)(1.06)*u(n*Min)= 1000 (which is the starting value)

Now if the problem asked for a different term (for example the 15th). You can either go to the table or since you have the equation entered you can put in *u*(the term) on the home screen and press enter and get the answer.

**Example Problem 2**

For the following problem:

- Tell what kind of sequence it is
- Write the next 2 terms
- Find t100
- Find the term number of the last given term

**2.** 2, 4, 8...32

Ok so this is an geometric sequence where the common ratio is 2. So to get to the next term you multiply the previous term by 2. The next two terms are 16 and 32. Now to find t100 you can figure out the explicit formula and either plug in 100 for the n value or you can plug in the formula into y= and go to the table at 100. The explicit formula formula for this sequence is Tn=2(2) ^(*n*-1) . T100 = 1.267 *10^30. Now to find the term number of the last given term you can either look for the term in the table or you can plug it in for Tn in the formula.

32= 2(2) ^(*n*-1)

16=(2) ^(*n*-1)

log 16 =( *n*-1) log 2

*n*= the 5th term

**Additional Help:**

http://home.alltel.net/okrebs/page131.html or http://www.purplemath.com/modules/series3.htm

this site is also kinda cool :http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=340

**Reminder: Natalie you are next!**

**Ok so...**our dance teacher told us to watch this clip. It's CRAZY. This woman is balancing and dancing in pointe shoes on this guy's head. Absolutely amazing (and crazy!) You have to watch it for a minute before she dances on his head...

## 0 Comments:

Post a Comment

<< Home