### 14-2 Arithmetic, Geometric, and other Sequences

**14.2 Arithmetic, Geometric, and other Sequences**

**Sequence**- is a function, with a term of pattern (n term - integer) and value (a number)

**Recursion formula**- specifies t

*n*as a function of the previous term (t

*n-1*)

**Explicit formula**- specifies t

*n*as a function of n.

**Arithmetic Sequence**- Add or Subtract the same number to each preceding term

**Common Difference**- the constant number being added or subtracted in an arithmetic sequence

**Arithmetic Linear Function**-

Explicit: T

*n*= T

*o*+ D(n-1)

Recursive: T

*n*= T

*n-1*+ D and To = x

where T

*n*is a value in the sequence, T

*o*is the first value in the sequence, D is the common difference and T

*n-1*is the previous term.

**Geometric Sequence**- Multiply or Divide the same number to each preceding term

**Common Ratio**- the constant number being multiplied or divided in a geometric sequence

**Geometric Exponential function**-

**Explicit**: T

*n*= T

*o** R(^n-1)

**Recursive**: T

*n*= T

*n-1**R and T

*o*= x

where T

*n*is a value in the sequence, T

*o*is the initial value in the sequence, T

*n-1*is the previous term and R is the common ratio.

**Sequence Mode on the Calculator -**

- MODE
- SEQ
- Y=
*n*Min = (this is the starting term number..normally 1)- u (
*n*) = (your equation goes here: u(n-1) + 5 ("u" can be found by 2ND 7 and "n" is the x, T, Theta, n button) - u (
*n*Min) = (this is your starting term value) - in order to find a certain term in a sequence go to TBLSET (2ND, WINDOW ) and make the TblStart whatever term you are looking for then go to TABLE (2ND GRAPH ) and the value will be in the table under u(
*n*).

**Example Problem: **

a. Tell Whether the sequence is arithemetic, geometric

b. Write the next 3 terms

c. Find t*76*

*d.* Find the term number of the term after the first ellipsis marks.

1. 26, 41.5, 57... ,6071.

2. 843, 140.5, 23.417... ,0.1087.

**HOW TO SOLVE:**

a. First you must look for a common difference or ratio by subtraction terms into each other like 41.5-26 = 15.5 and 57 - 41.5 = 15.5 so the common difference is add 15.5. If there is no common difference then you must divide terms into each other like 843 / 140.5 = 6 and 140.5 / 23.417 = about 6 because of rounding. So the common ratio is divide by 6. The first sequence is arithmetic because there is a common difference whereas the second sequence is geometric because there is a common ratio.

b. Use the seq mode TABLE with the TblStart = 1 in the TBLSET to see the next 3 terms.

c. Use the seq mode TABLE with the TbleStart = 76 in the TBLSET.

d. You can search for this value in the TABLE but it is much easier to solve for n in the equation for the sequence. So, since the first sequence is arithmetic use Tn = To + D(n-1) to solve for n: 6071 = 26 + 15.5 (n-1). So subtract 26 from both sides and then divide by 15.5 to both sides and 390 = n-1, then add one to both sides and n=391 meaning 6070 is the 391st term in the sequence. For geometric sequences, using the explicit formula to solve for n, you must use log to get the n out of the exponent. So Tn = To * R(^n-1) : 0.1087 = 843 / 6^(n-1). Divide 843 by 0.1087 (=7755.289). Then take the log of both sides which moves n-1 infront of log6. Then divide both sides by log 6. So log 7755.289 / log 6 = 4.999 = 5 = n-1. Then add one to both sides to get n= 6. So 0.1087 is the 6th term.

**ANSWERS**:

1. a. Arithmetic

b. 72.5, 88, 103.5

c. 1188.5 is the 76th term.

d. 6071 is the 391st term.

2. a. Geometric

b. 3.9028, 0.65046, 0.10841

c. 4 * 10 ^(-56)

d. 0.1087 is the 6th term.

**Additional Information** can be found here:

http://home.alltel.net/okrebs/page131.html

**Quiz yourself** at this neat website: http://www.ltcconline.net/greenl/java/IntermedCollegeAlgebra/ArithGeo/ArithGeo.html

**Madison** you are up next!!!

Trey Kozacik, Xander Berry and I did a science fair project and we won 2nd place in our division on Saturday in the LA County Science fair. This means that we are going to STATE!!! Who would've thought considering we did this project purely for fun and then were convinced by teachers to turn it into a science fair project!! Our project is called Shooting For Distance (I know, lame right..well I wanted something a little fun..I was going for SHOOTING FOR THE STARS....) We built a 10 ft air cannon that shoots weighted water bottles up to about 900 ft.