2.1 From Figures to Expressions
Here below you see a few figures. When we talk about sequences, we usually call every figure for element or term.
In every term, the number of stars is increased. How many stars do you think there will be in term 4 and term 5?
Most likely, you see that there should be 8 stars in term 4 and 10 stars in term 5.


Instead of drawing all 100 terms, we can find an expression for calculating the number of stars.
For every term, the number of stars increases by 2.
We then write up the following table: 
We indicate the number of stars in the term “n” for f(n).
The term’s placement in the sequence is called n.
The number of stars in term "n" are f(n) = 2 · n
In the hundredth term, it would be then f(100) = 2 · 100 = 200 stars.
This can also be shown with a graph in a coordinate system. We being by making a value table.
You see now clearly that the graph of the relation creates a straight line. Because of this, the graph is said to be a linear function.
Here below, you see the prices for two mobile phone subscriptions in the form of a table and as a graph in a coordinate system.

Foto: Fredrik Enander 


We can see in the table that company A that the price is proportional to the number of minutes you talk for if: 

 the number of minutes doubles, then the cost becomes twice as much.
1 min = 1 kr
2 min = 2 kr
 the number of minutes becomes half as large, then the costs is half the size.
4 min = 4 kr
2 min = 2 kr
We can see that in company B that the price is not proportional to the number of minutes you talk if:
 the number of minutes doubles, then the cost is not twice as large.
1 min = 1.50 kr
2 min = 2 kr
 the number of minutes becomes half as large, then the cost does not become half as large.
4 min = 3 kr
2 min = 2 kr

We can see this now when both company’s prices are in a coordinate system.


