3.2 Patterns and Algebraic Expressions When figures are constructed according to a pattern, we can express this pattern as an algebraic expression. This means that we can reconstruct any figure with a given pattern by using an algebraic expression. 

Example 1
How many matches does the figure n have? 

Figure 1 has in the example above 4 matches. Figure 2 has 7 matches. Figure 3 has 10 matches and so on. It is more practical to show this in a table form in order to get an overview and then be able to create an algebraic expression for how many matches n has. 

We can see that the number of matches increases by 3 in every figure. This can even be written in the follow way: 
Figure 1: 
3 · 1 + 1 = 4 matches 
Figure 2: 
3 · 2 + 1 = 7 matches 
Figure 3: 
3 · 3 + 1 = 10 matches 
Figure 4: 
3 · 4 + 1 = 13 matches 
Figure 5: 
3 · 5+ 1 = 16 matches 
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Figure n: 
3 · n+ 1 = ? matches 

If we want to calculate the number of matches in figure 14 in the order, we replace the variable n with 14:
Figure 14 has 3 · 14 + 1 = 43 matches 

Example 2
How many black squares does figure n have? 

We begin by recreating the number of black squares in table form. By doing so, we can more easily find a pattern. 

Now, we need to find a pattern for how many black square each figure increases by. We can see that the whole time each figure is squared:
1 · 1 = 1
2 · 2 = 4
3 · 3 = 9
and so forth
It seems though that it is moved one step to the right.

Figure 1: 
(1  1)^{2} = 0^{2} = 0 black squares 
Figure 2: 
(2  1)^{2} = 1^{2} = 1 black squares 
Figure 3: 
(3  1)^{2} = 2^{2} = 4 black squares 
Figure 4: 
(4  1)^{2} = 3^{2} = 9 black squares 
Figure 5: 
(5  1)^{2} = 4^{2} = 16 black squares 
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Figure n: 
(n  1)^{2} = ? black squares 

Answer: In the figure n, the numbers of black squares is (n  1)^{2}.
How many black squares are there in figure 9 in the order?
We replace the variable n with 9:
Figure 9 has (9  1)^{2} = 8^{2} = 64 black squares 



