Chapter 1

Mathematical Reasoning

Section 1.4

Patterns

Patterns are common to every area of study and are found in both man made objects and in nature. Many patterns develop in the study mathematics in arithmetic and geometry. The problem is that in mathematics patterns are explained or described with great detail. Discovering and extending patterns requires both inductive and deductive reasoning. In this section we will talk about certain number patterns and show how they can explain some other patterns in geometry for example.

Number Sequence Patterns

Find the next three terms in each of the following number sequences.

Sequence 1: 2, 5, 8, 11, 14, 17, 20, , , ,…,77,

(Keep adding 3 “Arithmetic Sequence”)

Sequence 2: 25, 250, 2500, , ,…, 2500000,…

(Keep Multiplying by 10 “Geometric Sequence”)

Sequence 3: 100, 88, 76, 64, , , ,…,4, ,…

(Keep subtracting 12 “Arithmetic Sequence”)

23 26 29 80

25000 250000

52 40 28 -8

The arithmetic sequence and geometric sequence are two of the most well-known number sequence patterns in mathematics. We will give some more details about each one.

Arithmetic Sequences

The next number in an arithmetic sequence can be found by adding (or subtracting) the same number from the current number in the sequence. The number you start with this book calls the INITIAL value. You can get your CURRENT value by adding a certain number to the PREVIOUS value.

For Example

In Sequence 1 (previous slide): 2, 5, 8, 11, 14, 17, 20, 23,…

Initial=2 and CURRENT = PREVIOUS + 3

Using the INITIAL value and the amount you increase or decrease by we can get a formula for the sequence.

n 1 2 3 4 5 6 72+3(n-1) 2 5 8 11 14 17 20

Let try this for Sequence 3:100, 88, 76, 64, 52,…

INITIAL =

CURRENT = PREVIOUS

The formula can be seen in the following table:

100

- 12

n 1 2 3 4 5 6 7100-12(n-1) 100 88 76 64 52 40 28

For an arithmetic sequence with INITIAL = a and CURRENT=PREVIOUS + d we get the following formula:

n 1 2 3 4 5 6 7ad(n-1) a ad a2

da3

da4

da5

da6

d

Use a + when the same number is added and a – when it is subtracted.

Geometric Sequences

Sequence 2: 25, 250, 2500, 25000, 250000,…

This can also be written using the INITIAL, CURRENT and PREVIOUS values in the following way:

INITIAL = 25 and CURRENT = PREVIOUS 10

n 1 2 3 4 5

25 250 2500 25000 250000)1(1025 n

In general with INITIAL = a and CURRENT = PREVIOUS r we get the following:

n 1 2 3 4 5)1( nra a ra 2ra 3ra 4ra

AREA PERIMETER

The is no one set method that works for finding patterns. If there where that would be what we would teach you to do. The challenge is how can you apply the techniques of inductive and deductive reasoning to begin to find a pattern.

Example

In the example below the squares are of length 1 on each side. I want to find the area (number of squares) and perimeter (distance around the outside) of each pattern.

1

2

3

4

4

6

8

10

AREA: INITIAL = 1 and CURRENT = PREVIOUS + 1

PERIMETER: INITIAL = 4 and CURRENT = PREVIOUS + 2

This will give the following pattern for the nth shape.

AREA of nth shape = 1+1(n-1)=1+n-1=n

PERIMETER of the nth shape = 4+2(n-1)=4+2n-2=2+2n

If the area is 7 what is the perimeter?

If the perimeter is 42 what is the area?

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