In earlier grades you saw patterns in the form of pictures and numbers. In this chapter, we learn more about the mathematics of patterns. Patterns are recognisable as repetitive sequences and can be found in nature, shapes, events, sets of numbers and almost everywhere you care to look. For example, seeds in a sunflower, snowflakes, geometric designs on quilts or tiles, the number sequence 0;4;8;12;16;....
Can you spot any patterns in the following lists of numbers?
2;4;6;8;10;...
1;2;4;7;11;...
1;4;9;16;25;...
5;10;20;40;80;...
Numbers can have interesting patterns. Here we list the most common patterns and how they are made.
Examples:
1;4;7;10;13;16;19;22;25;... This sequence has a difference of 3 between each number. The pattern is continued by adding 3 to the last number each time.
3;8;13;18;23;28;33;38;... This sequence has a difference of 5 between each number. The pattern is continued by adding 5 to the last number each time.
2;4;8;16;32;64;128;256;... This sequence has a factor of 2 between each number. The pattern is continued by multiplying the last number by 2 each time.
3;9;27;81;243;729;2187;... This sequence has a factor of 3 between each number. The pattern is continued by multiplying the last number by 3 each time.
1 ; 3 ; 6 ; 10 ; 15 ; 21 ; 28 ; 36 ; 45 ; . . .
This sequence is generated from a pattern of dots which form a triangle. By adding another row of dots (with one more dot in each row than in the previous row) and counting all the dots, we can find the next number of the sequence.
1 ; 4 ; 9 ; 16 ; 25 ; 36 ; 49 ; 64 ; 81 ; . . .
The next number is made by squaring the number of the position in the pattern.
The second number is 2 squared (
).
The seventh number is 7 squared (
) etc.
1 ; 8 ; 27 ; 64 ; 125 ; 216 ; 343 ; 512 ; 729 ; . . .
The next number is made by cubing the number of the position in the pattern.
The second number is 2 cubed (
).
The seventh number is 7 cubed (
) etc.
0 ; 1 ; 1 ; 2 ; 3 ; 5 ; 8 ; 13 ; 21 ; 34 ; . . .
The next number is found by adding the two numbers before it together. The 2 is found by adding the two numbers in front of it (1+1). The 21 is found by adding the two numbers in front of it (8+13). The next number in the sequence above would be 55 (21+34).
Can you figure out the next few numbers?
Say you and 3 friends decide to study for Maths, and you are seated at a square table. A few minutes later, 2 other friends join you and would like to sit at your table and help you study. Naturally, you move another table and add it to the existing one. Now 6 of you sit at the table. Another 2 of your friends join your table, and you take a third table and add it to the existing tables. Now 8 of you can sit comfortably.
Examine how the number of people sitting is related to the number of tables.
Tabulate a few terms to see if there is a pattern :
| Number of Tables, n | Number of people seated |
| 1 | 4 = 4 |
| 2 | 4 + 2 = 6 |
| 3 | 4 + 2 + 2 = 8 |
| 4 | 4 + 2 + 2 + 2 = 10 |
| ⋮ | ⋮ |
| n | 4 + 2 + 2 + 2 + ... + 2 |
We can see that for 3 tables we can seat 8 people, for 4 tables we can seat 10 people and so on. We started out with 4 people and added two each time. Thus, for each table added, the number of persons increased by 2.
The nth-term of a sequence is written as an. So for example, the 1st-term of a sequence is a1, the 10th-term is a10.
A sequence does not have to follow a pattern but when it does, we can often write down a formula to calculate the nth-term, an. In the sequence
where the sequence consists of the squares of integers, the formula for the nth-term is
You can check this by looking at:
Therefore, using Equation 3.2, we can generate a pattern, namely squares of integers.
We can also define the common difference for a pattern.
The common difference is the difference between successive terms and is denoted by d.
For example, consider the sequence 10;7;4;1;... . To find the common difference, we simply subtract each successive term:
As before, you and 3 friends are studying for Maths, and you are seated at a square table. A few minutes later, 2 other friends join you and add another table to the existing one. Now 6 of you can sit together. A short time later 2 more of your friends join your table, and you add a third table to the existing tables. Now 8 of you can sit comfortably as shown:
Find the expression for the number of people seated at n tables. Then, use the general formula to determine how many people can sit around 12 tables and how many tables are needed for 20 people.
Tabulate a few terms to see if there is a pattern :
| Number of Tables, n | Number of people seated | Formula |
| 1 | 4 = 4 | = 4 + 2 · ( 0 ) |
| 2 | 4 + 2 = 6 | = 4 + 2 · ( 1 ) |
| 3 | 4 + 2 + 2 = 8 | = 4 + 2 · ( 2 ) |
| 4 | 4 + 2 + 2 + 2 = 10 | = 4 + 2 · ( 3 ) |
| ⋮ | ⋮ | ⋮ |
| n | 4 + 2 + 2 + 2 + ... + 2 | = 4 + 2 · ( n – 1 ) |
The number of people seated at n tables is:
Notice how we have used d to represent the common difference. We could also have written a 2 in place of the d. We also used a1 to represent the first term, rather than using the actual value (4).
Considering the example from the previous section, how many people can sit around say 12 tables? We are looking for a12, that is, where n=12:
Calculate the number of terms if an=20 :
26 people can be seated at 12 tables and 9 tables are needed to seat 20 people.
It is also important to note the difference between n and an. n can be compared to a place holder, while an is the value at the place “held” by n. Like our “Study Table” example above, the first table (Table 1) holds 4 people. Thus, at place n=1, the value of a1=4 and so on:
| n | 1 | 2 | 3 | 4 | ... |
| an | 4 | 6 | 8 | 10 | ... |
Find the general formula for the following sequences and then find a10, a50 and a100: