1. Arithmetic Sequences and Geometric Sequences

2. Arithmetic Sequences
An arithmetic sequence is a set of numbers put into a specific order by a pattern of addition or subtraction.
an = a1 + (n – 1)d– This is the formula.
an represents the nth term, the unknown term that you are trying to find, of a sequence.
a1 is the first term in a sequence.
n is an unknown term that is always the same number as the n term in an.

3. Arithmetic Sequences (continued)
an=a1+(n-1)d
The d in the formula is the
Common Difference between
each of the terms in a series.
For example: 1, 5, 9, 13… The common difference (d) is +4.
The d term can also be negative:
10, 7, 4, 1, -2… The d term is -3
(this means that instead of
adding a number you
subtract it.)

4. Geometric Sequences an = a1rn-1 Geometric Sequence formula.
an is the unknown term (just like the arithmetic sequences)
a1 is the first term.
r is the rate, also known as the common ratio. It is the change between two terms in a geometric sequence. It is either a number being multiplied or divided. You can also multiply by (1) over the number being multiplied.

5. More Geometric Sequences
Some examples of geometric sequences are:
1, 2, 4, 8, 16, 32…-- r = 2
100, 50, 25, 12.5, 6.25…-- r = 1/2 (divide the preceding number by 2.)
an=a1rn-1

6. Some Interesting Example Equations
Geometric example: find the nth term.
a1 = -10, r=4, n=2
an = -10(4)2-1
an = -10(4)1
an = -40
Arithmetic example: find a14, a1=4, d=6
a14= 4 + (14-1)6
a14= a14= 82

7. How this relates to Real Life Outside Math Class
A painter is a job that requires the use of an arithmetic sequence to correctly space the things he is painting. If the painter was painting stripes on a wall, he could find the places to put the stripes to evenly space them.

8. Another Real Life Slide
If an owner of a store needed to count up the amount of stuff they sell, or how much money they make, he could use and arithmetic or geometric sequence.
If the owner had a pattern of how much money they make as time progresses, that is a sequence. The owner also needs these sequences if he/she wants to predict the earnings of his or her store in years to come.

9. Finished!