Chapter 4-1 Notes

Definitions

• Inductive Reasoning

• Making conclusions based upon observations and patterns

Examples

• Example 1

A dot pattern is shown below. How many dots would there be in the bottom row of the 4th figure? What would the total number of dots

be in the 6th figure?

4 dots on the bottom row of the 4th figure

21 total dots in the 6th

figure

Examples

• Example 2

How many triangles would be in the 10th figure? If one of these figures contained 34 triangles, how many squares would be in that

figure? If we knew the figure number, how could we find the number of triangles?

22 triangles in the 10th figure

If the figure had 34 triangles, it also has 16 squares

2n + 2, n is the figure number

Examples

• Example 3

Look at the pattern 2, 4, 6, 8, 10, … What is the 19th term in the pattern? Describe the pattern and find an equation that works for

every term in the pattern.

The 19th term is 38

2n, where n is the term number

Process

• Writing Number Pattern Equations

• Find the differences between

• If the difference is the , write times n

• If the difference is not the , find the “ “

• Plug 1 in for n, then subtract that value with in the pattern

• Subtract this value from

each term

same the difference

same second differences

the first term

your equation

Examples

• Example 4

Look at the pattern 1, 3, 5, 7, 9, 11, … What is the number pattern equation? What is the 34th term in the pattern?

1, 3, 5, 7, 9, 11, …

2, 2, 2, 2, 2, 2n

2(1) = 2

2 – 1 = 1

– 1

2(34) – 168 – 1

67

Examples

• Example 5

Look at the pattern 1, 4, 9, 16, 25, …. What is the number pattern equation for this pattern?

1, 4, 9, 16, 25, …3, 5, 7, 9,

2, 2, 2,𝒏𝟐

(𝟏)𝟐= 1

1 – 1 = 0

Examples

• Example 6

Find the number pattern equation and the 8th term for the following

pattern: 𝟐

𝟏,𝟑

𝟒,𝟒

𝟗,𝟓

𝟏𝟔,𝟔

𝟐𝟓, …

2, 3, 4, 5, 6, …1, 1, 1, 1

n

1 – 2

–1

+ 1

1, 4, 9, 16, 25, …

𝒏𝟐

𝒏 + 𝟏

𝒏𝟐

𝟖 + 𝟏

𝟖𝟐𝟗

𝟔𝟒

Definitions

• Conjecture

• An that is based on in a pattern

• Examples can make you believe a conjecture, but can never it

educated guess examples

prove

Examples

• Example 7

Here is an algebraic equation and a table of values for n with the result for t. after looking at the table, Pablo makes this conjecture: the value of (n-1)(n-2)(n-3) is 0 for any whole number value of n. Is

this a valid, or true, conjecture?

No, this is not a valid conjecture

Definitions

• Counterexample

• An example that a conjecturedisproves

Examples

• Example 8

Give a counterexample to Example 7.

If n = 4

(4-1)(4-2)(4-3)

(3)(2)(1)

(6)

•HW #21: 2, 4, 5-8, 10-11, 16-20, 26 (E.C.)