11.1 Mathematical Patterns11.1 Mathematical Patterns

Ex 1Ex 1

Start with a square with sides 1 unit Start with a square with sides 1 unit long. On the right side, add on a long. On the right side, add on a square of the same size. Continue square of the same size. Continue adding one square at a time in this adding one square at a time in this way. Draw the first four figures.way. Draw the first four figures.

Ex 2Ex 2

Write the number of 1 unit segments Write the number of 1 unit segments in each figure from ex 1 as a in each figure from ex 1 as a sequence.sequence.

Ex 3Ex 3

Describe the pattern formed and find Describe the pattern formed and find the next three terms. 243, 81, 27, 9, the next three terms. 243, 81, 27, 9, ……

Ex 4Ex 4

Suppose you drop a ball from a Suppose you drop a ball from a height of 100 cm. It bounces back to height of 100 cm. It bounces back to 80% of its previous height. How high 80% of its previous height. How high will it go after its fifth bounce?will it go after its fifth bounce?

We can use a variable with positive integer We can use a variable with positive integer subscripts to represent the terms in a subscripts to represent the terms in a sequence:sequence:

aa11 aa22 aa33 – first, second and third terms – first, second and third terms

aan-1:n-1: n – 1 term n – 1 term

aan :n : nth term nth term

aan+1n+1 :: n + 1 term n + 1 term

n is the term numbern is the term number

Recursive formulaRecursive formula

Defines the terms in a sequence by Defines the terms in a sequence by relating each term to the ones before relating each term to the ones before it. (ex 4 was recursive b/c the height it. (ex 4 was recursive b/c the height was 80% of its previous height)was 80% of its previous height)

Formula would be aFormula would be ann = 0.80a = 0.80an-1n-1 where where

aa11 = 100 = 100

Ex 5Ex 5

Describe the pattern of the Describe the pattern of the sequence: sequence:

2, 6, 18, 54, 162, …2, 6, 18, 54, 162, …

Write a recursive function.Write a recursive function.

Find the 6Find the 6thth and 7 and 7thth terms. terms.

Find the value of aFind the value of a1010

Explicit formulaExplicit formula

Expresses the nth term in terms of nExpresses the nth term in terms of n

Finding the value of a term without Finding the value of a term without knowing the preceding term.knowing the preceding term.

(Find a link between the term (Find a link between the term number and the term value.)number and the term value.)

EX 6EX 6Write a formula.Write a formula.

2, 6, 12, 20, …2, 6, 12, 20, …

Ex 7Ex 7Write a formulaWrite a formula

3, 5, 7, 9, …3, 5, 7, 9, …

Ex 8Ex 8TermTerm aa11 aa22 aa33 aa44

Length of sideLength of side 11 22 33 44 perimeterperimeter 55 1010 1515 2020

For each sequence, find the next term and For each sequence, find the next term and the 20the 20thth term. term.

Write an explicit formula for each sequence. Write an explicit formula for each sequence.

n = the term numbern = the term number

Ex 9Ex 9

Write the first six terms of the area Write the first six terms of the area of squares that have side lengths 1, of squares that have side lengths 1, 2, 3, etc.2, 3, etc.

Write an explicit formula.Write an explicit formula.

Ex 10Ex 10Write a formula for Write a formula for 1 1 1

1, , , ,...3 9 27

Ex 11Ex 11Write a formula for: Write a formula for:

1 1 11, , , ,...2 3 4