Algebra 1-2: Sequences - Geometric and Arithmetic

Zombies. Yes, zombies. Below are the two data sets for the zombie virus cases in the cities that are reporting sequential data.

London		Tokyo
Day	Incidence Report	Day	Incidence Report
1	3	1	4
2	4.5	2	8
3	6.75	3	12
4	10.125	4	16
5	15.1875	5	20
6	22.78125	6	24
7	34.171875	7	28
8	51.2578125	8	32
9	76.88671875	9	36
10	115.3300781	10	40
11	172.9951172	11	44
12	259.4926758	12	48
13	389.2390137	13	52
14	583.8585205	14	56

Do I even have a sequence?
When we're working with sequences, we first need to determine if it is actually a sequence. This step is covered if you can find a pattern in the data. In other words, what is your rule? If you can't find a rule to get from one data point to the next, then you don't have a sequence.

Arithmetic Sequences
Next you need to determine if the sequence is arithmetic or geometric. Arithmetic sequences have a rule that deals with addition (think basic "arithmetic" is adding...). In our zombie case, we have the data set from Tokyo that has a rule with addition. Specifically, our rule is to "+4", meaning we add 4 to a data point to get the next data point.
We can also plot the data on a graph. If your graph shows a linear function, and you can find that rule, you're going to have an arithmetic sequence. Below is a picture of the graph of our Tokyo data.

We can use our rule to find other terms in the sequence, but if we wanted to know how many people would be infected with the zombie virus after 30 days, that would be a lot of computation. Instead, we can use the general formula for arithmetic sequences and tweak it to fit our data.

General Formula for Arithmetic Sequences: y = a + (x - 1)d
a = initial Amount
x = the value of the independent variable we want to measure (for example, if we're looking at the Tokyo data and want to find how many will be infected after 30 days, our x = 30)
d = the common difference (this is your rule - what are you adding to get to the next number?)

In the case of poor Tokyo, there were initially 4 people infected. This makes our a = 4. We also know that our rule is to "+4". This makes our d = 4. Guess what? We have everything we need to make the general equation fit our data.

y = a + (x - 1)d

y = 4 + (x - 1)4

But does this equation actually work? Below is the graph of our Tokyo data with that exact equation superimposed over it.

Pretty decent fit, wouldn't you say?

Geometric Sequences

Geometric sequences have a rule that deals with multiplication (something we do a lot of in geometry...). In our zombie case, we have the data set from London that has a rule with multiplication. Specifically, our rule is to "times 1.5", meaning we take a data point and multiply its value by 1.5 to get the next data point.

We can also plot the data on a graph. If your graph shows a curved function, and you can find that rule, you're going to have an geometric sequence. Below is a picture of the graph of our London data.

General Formula for Geometric Sequences: y = a * r^(x - 1)

a = initial Amount

x = the value of the independent variable we want to measure (for example, if we're looking at the Tokyo data and want to find how many will be infected after 30 days, our x = 30)

r = the common ratio (this is your rule - what are you multiplying by to get to the next number?)

For London, there were initially 3 people infected. This makes our a = 3. We also know that our rule is to "times 1.5". This makes our r = 1.5. Guess what? We have everything we need to make the general equation fit our data.

y = a * r^(x - 1)

y = 3 * 1.5^(x - 1)

But does this equation actually work? Check out the graph of our London data with that exact equation superimposed over it: