Introduction

Number patterns - grade 10

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,....

Investigation : patterns

Can you spot any patterns in the following lists of numbers?

1. 2; 4; 6; 8; 10; $...$
2. 1; 2; 4; 7; 11; $...$
3. 1; 4; 9; 16; 25; $...$
4. 5; 10; 20; 40; 80; $...$

Common number patterns

Numbers can have interesting patterns. Here we list the most common patterns and how they are made.

Examples:

1. $1,4,7,10,13,16,19,22,25,...$ This sequence has a difference of 3 between each number. The pattern is continued by adding 3 to the last number each time.
2. $3,8,13,18,23,28,33,38,...$ This sequence has a difference of 5 between each number. The pattern is continued by adding 5 to the last number each time.
3. $2,4,8,16,32,64,128,256,...$ This sequence has a factor of 2 between each number. The pattern is continued by multiplying the last number by 2 each time.
4. $3,9,27,81,243,729,2187,...$ This sequence has a factor of 3 between each number. The pattern is continued by multiplying the last number by 3 each time.

Special sequences

Triangular numbers

$1,3,6,10,15,21,28,36,45,...$

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.

Square numbers

$1,4,9,16,25,36,49,64,81,...$

The next number is made by squaring the number of the position in the pattern. The second number is 2 squared ( $2^{2}or2\times 2$ ). The seventh number is 7 squared ( $7^{2}or7\times 7$ ) etc.

Cube numbers

$1,8,27,64,125,216,343,512,729,...$

The next number is made by cubing the number of the position in the pattern. The second number is 2 cubed ( $2^{3}or2\times 2\times 2$ ). The seventh number is 7 cubed ( $7^{3}or7\times 7\times 7$ ) etc.

Fibonacci numbers

$0,1,1,2,3,5,8,13,21,34,...$

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.

1. 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	$4+2+2+2=10$
   $⋮$	$⋮$
   $n$	$4+2+2+2+...+2$

2. Describe the pattern

   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.