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Warm Up

Unit 4: Discrete Functions

Unit 4: Discrete Functions

Discrete Functions covers topics on patterns, arithmetic and geometric sequences and series and then moves into financial math.

Financial Applications: Simple interest, compound interest, present value, amount of an annuity and present value of an annuity are all covered as applications of financial math.

Lesson 1: Sequences

Learning Goals & Success Criteria

• Identify and Classify Sequences

• Create functions for describing sequences and use the sequences to make predictions

What does the term “sequence” mean in everyday language?

What is the most important aspect of a sequence?

Sequence

• In mathematics, a sequence is a set of numbers, usually separated by commas, arranged in a particular order.

Sequences in Our Lives

• Many natural phenomena, such as the spiral patterns seen in seashells, sunflowers, and galaxies, can be represent by sequences

Ex. “The Golden Ratio”

“Beauty is in the phi of the beholder”

Why does it matter to us in MCR3U?

• Some sequences have very specific patterns and can be represented by mathematical rules or functions.

Definition: Sequence

• A set of numbers arranged in order.

• This set is identified by a pattern or rule that may stop at some number or it may continue indefinitely.

• Ex. 3, 7, 11, 15,

• Ex. 2, 6, 18, 54, …

Definition: Sequence

• A function is considered to be a sequence if it’s domain is the set, or a subset, of the natural numbers (positive whole integers) and whose range is the terms of a sequence

Definition: Term

• a single value (number) or object in a sequence is a term.• Ex. 3, 12, 21, 30, …

• Subscripts are usually used to identify the positions of the terms.• In the example above,

• 𝑡𝑒𝑟𝑚 1 = 𝑡1 = 3

• 𝑡𝑒𝑟𝑚 2 = 𝑡2 = 12

• 𝑡𝑒𝑟𝑚 3 = 𝑡3 = 21

• 𝑡𝑒𝑟𝑚 4 = 𝑡4 = 30

• 𝑡𝑒𝑟𝑚 5 = 𝑡5 = _______

Definition: General Term

• A formula, labelled, 𝑡𝑛, that expresses each term of a sequence as a function of its position.

• Example:

If the general term is 𝑡𝑛 = 2𝑛,

then to calculate the 12th term 𝑡12 , we would substitute 𝑛 = 12 into our general term

𝑡𝑛 = 2𝑛⇒ 𝑡12 = 2 12

𝑡12 = 24

Definition: Explicit Formula

• A formula that represents any term in a sequence relative to the term number, 𝑛, where 𝑛 ∈ ℕ (1, 2, 3,…)

Example 1: Use the Explicit Formula to Write TermsWrite the first three terms of each sequence, given the explicit formula for the 𝑛𝑡ℎ term of the sequence 𝑛 ∈ ℕ .

a)𝑡𝑛 = 3𝑛2 − 1

b)𝑡𝑛 =𝑛−1

𝑛

Example 2: Determine Explicit Formulae in Function Notation

For each sequence,

i. Make a table of values using the term number and term.

ii. Calculate the finite differences. (first, second, … until its constant)

iii. Graph the sequence using the ordered pairs from PART i.

iv. Determine an explicit formula for the nth term, using function notation.

a) 7, 12, 17, 22, …b) 1, 10, 25, 46

Sequences

Arithmetic Geometric+/− ×

Definition: Arithmetic Sequence

• A sequence that has the same difference, common difference (𝑑), between any pair of consecutive terms.

Definition: Explicit Formula

• An explicit formula for the nth term of a sequence can sometimes be determined by finding a pattern among the terms.

Term Number Term

1 𝑎

2 𝑎 + 𝑑

3 𝑎 + 𝑑 + 𝑑 = ________________

4

5

6

… …

Definition: Explicit Formula

An arithmetic sequence can be written as 𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑,…

where 𝒂 is the first term and 𝒅 is the common difference. Then the formula for the general term, or the 𝑛𝑡ℎ term, of an arithmetic sequence is

𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑,𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ ℕ

Definition: Explicit FormulaExpanding & simplifying the right side of the equation allows to rewrite

our General Term Explicit Formula as a FUNCTION!

𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑,𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ ℕ

𝑡𝑛 = 𝑎 + 𝑑 ∙ 𝑛 − 𝑑,𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ ℕ

𝑡𝑛 = 𝑑 ∙ 𝑛 + (−𝑑 + 𝑎), 𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ ℕ

Comparable to 𝒇 𝒙 = 𝒎𝒙 + 𝒃, with the restriction 𝑥 ∈ ℕ

𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒

𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡(𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑜𝑟 #) 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡

𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒

𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑒𝑟𝑚

So…

• We can see that arithmetic differences have a LINEAR RELATIONSHIP.

• So we that the arithmetic function must be a linear function

Example 3:

How many terms are there in the following sequence?

1, 4, 7, …, 121

Example 4:In the arithmetic sequence, 𝑡5 = 16 𝑎𝑛𝑑 𝑡8 = 25. Find 𝑡1(𝑎) and d, 𝑓𝑜𝑟 𝑡𝑛.

16, ___, ___, 25𝑡5 𝑡6 𝑡7 𝑡8

𝑡8−𝑡5= 25-163 differences = 9

3d = 93d/3 = 9/3

d = 3

𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑,𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ ℕ

If 𝑡5 = 16 & d = 3

⇒ 𝑡𝑛 = 𝑎 + 𝑛 − 1 3⇒ 𝑡5 = 𝑎 + 5 − 1 3⇒ 16 = 𝑎 + 4 316 = 𝑎 + 1216 − 12 = 𝑎

𝑎 = 4

∴ 𝒕𝒏 = 𝟒 + 𝒏 − 𝟏 𝟑, 𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ ℕ

Example 5: Length of Ownership

Anna paid $500 for an antique guitar. The guitar appreciates in value by $160 every year. If she sells the guitar for a little over $7000, how long has she owned it?

Sequences

Arithmetic Geometric+/− ×/÷

Definition: Geometric Sequences

• A sequence where the ratio, the common ratio, of consecutive terms is a constant.• What operation does ratio refer to?

• Common Ratio: refers to the ratio of any two consecutive terms in a geometric sequence (GS).

Geometric Sequence

The terms of a GS are obtained by multiplying the first term, 𝑎, and each subsequent term by a common ratio, 𝑟.

A GS can be written as 𝑎, 𝑎𝑟, 𝑎𝑟2, 𝑎𝑟3, 𝑎𝑟4,…

Then the formula for the general term or the 𝑛𝑡ℎ term of a GS is

𝑡𝑛 = 𝑎𝑟𝑛−1 𝑤ℎ𝑒𝑟𝑒 𝑟 ≠ 0 𝑎𝑛𝑑 𝑛 ∈ ℕ

Geometric Sequence – The Explicit FormulaRewriting the right side of the equation in terms of 𝑥 & 𝑓 𝑥 allows to

rewrite our General Term Explicit Formula as

a FUNCTION! (an EXPONENTIAL FUNCTION)

𝑡𝑛 = 𝑎𝑟𝑛−1 𝑤ℎ𝑒𝑟𝑒 𝑟 ≠ 0 𝑎𝑛𝑑 𝑛 ∈ ℕ

Comparable to 𝒇 𝒙 = 𝑎𝑏𝑥−1, with the restriction 𝑥 ∈ ℕ

𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒

𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡(𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑜𝑟 #)

𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒

𝑏𝑎𝑠𝑒

Example 1: Determine the Type of Sequence

Determine whether each sequence is arithmetic, geometric, or neither. Justify your answer.

a) 2, 5, 10, 17,…

b) 0.2, 0.02, 0.002, 0.0002,…

c)𝑎 + 2, 𝑎 + 4, 𝑎 + 6, 𝑎 + 8,…

Example 1: a) 2, 5, 10, 17,…

1. Check if the sequence is arithmetic (i.e., is there a common first difference?)

∴ as there is no common first difference,

we can conclude that this is not an

arithmetic sequence

Term #, n Term, 𝒕𝒏

1 2

2 5

3 10

4 17

First Difference

5 – 2 = 3

10 – 5 = 5

17 – 10 = 7

Example 1: a) 2, 5, 10, 17,…

2. Check if the sequence is geometric (i.e., is there a common first ratio?)

∴ as there is no common ratio,we can conclude that this is

not a Geometric Sequence

Term #, n Term, 𝒕𝒏

1 2

2 5

3 10

4 17

First Ratio

5

2= 2.5

10

5= 2

17

10= 1.7

a) 2, 5, 10, 17

b) 0.2, 0.02, 0.002, 0.0002,…

c)𝑎 + 2, 𝑎 + 4, 𝑎 + 6, 𝑎 + 8,…

Term #, n Term, 𝒕𝒏

1

2

3

4

First Difference Term #, n Term, 𝒕𝒏

1

2

3

4

First Ratio

Example 2: Write the terms in a GS

Write the first three terms of each GS.

a) 𝑓 𝑛 = 5(3)𝑛−1

b) 𝑡𝑛 = 161

4

𝑛−1

c) 𝑎 = 125 𝑎𝑛𝑑 𝑟 = −2

𝑡𝑛 = 𝑎𝑟𝑛−1 𝑤ℎ𝑒𝑟𝑒 𝑟 ≠ 0 𝑎𝑛𝑑 𝑛 ∈ ℕ

Example 3: Determine the number of terms

Determine the number of terms in the GS

4, 12, 36, …, 2916

𝑡𝑛 = 𝑎𝑟𝑛−1 𝑤ℎ𝑒𝑟𝑒 𝑟 ≠ 0 𝑎𝑛𝑑 𝑛 ∈ ℕ

Example 4:

Seatbelt use became law in Canada in 1976. Since that time, the number of deaths due to motor vehicle collisions has decreased. From 1984 to 2003, the number of deaths decreased by about 8% every 5 years. The number of deaths due to motor vehicle collisions in Canada in 1984 was approximately 4100.

a) Determine a formula to predict the number of deaths for any fifth year following 1984.

b) Write the number of deaths as a sequence for five 5-year intervals.

𝑡𝑛 = 𝑎𝑟𝑛−1 𝑤ℎ𝑒𝑟𝑒 𝑟 ≠ 0 𝑎𝑛𝑑 𝑛 ∈ ℕ

Homework

• Pg. 424• #1 - 4, 8 - 10, 12, 14 - 17

• Pg. 430• #2 - 4, 6, 8, 9 - 11, 13, 15, 16, 17, 18, 20 - 22.