DO NOW:

Unit 1 Sequences (Ch. 1 ) & Series (Ch. 9)F 502. Find the next term in a sequence described recursivelyF603. Find a recursive expression for the general term in a sequence described recursivelyAF 701. Solve complex arithmetic problems involving percent of increase or decrease or requiring integration of several concepts (e.g., using several ratios, comparing percentages, or comparing averages) F 703. Exhibit knowledge of geometric sequences

Ch. 1.1-1.5 Objectives:

Use recursive formulas for generating arithmetic, geometric & shifted geometric sequences.

Recognize arithmetic & geometric sequences from their graphs.

Use geometric sequences to model growth & decay.

Explore long-run values of geometric & shifted geometric sequences. Use graphs to check whether a recursive formula is a good model for data. Use shifted geometric sequences to model loans & investments.

Alg. II / Trig. Unit 1 Sequences (Ch. 1 ) & Series (Ch. 9) NAME: _______________

ACT Standards Block: __-__

F 502. Find the next term in a sequence described recursivelyF 603. Find a recursive expression for the general term in a sequence described recursivelyAF 701. Solve complex arithmetic problems involving percent of increase or decrease or requiring integration of several concepts (e.g., using several ratios, comparing percentages, or comparing averages)F 703. Exhibit knowledge of geometric sequences

I know I am successful when I can:

Use recursive formulas for generating arithmetic, geometric & shifted geometric sequences.Recognize arithmetic & geometric sequences from their graphs.Use geometric sequences to model growth & decay.Explore long-run values of geometric & shifted geometric sequences.Use graphs to check whether a recursive formula is a good model for data.Use shifted geometric sequences to model loans & investments.

Lesson Checklist:___ 1.1 Recursively Defined Sequences #(1-5)

___ 1.2 Modeling Growth & Decay #(1-5)

___ 1.3 A First Look at Limits #(1-4)

___ 1.4 Graphing Sequences #(1-3)

___ 1.5 Loans & Investments #(1-4)

Ch. 1.1 Recursive SequencesLearning Intentions:

Discover recursive formulas for sequences.

Define, explore & use arithmetic & geometric sequences.

Use recursively defined sequences to model real-life situations.

Recursion: applying a procedure repeatedly, starting with a number orgeometric figure, to produce a sequence of numbers or figures.~ Each term or stage builds on the previous term or stage.

u = the value or output of each term n = the location (instance) of each term

Sequence: an ordered list of numbers. {𝒖𝟏, 𝒖𝟐, 𝒖𝟑, 𝒖𝟒, 𝒖𝟓, …𝒖𝒏}

Term: each number in the sequence (each 𝒖 – value)

1st term / starting value: 𝒖𝟏 ‘u sub 1’ (𝒖𝟐 = 2nd term…)

Previous term: 𝒖𝒏−𝟏

ex.) If n = 7, then 𝒖𝟕 is the 7th term & 𝒖𝟕−𝟏 or 𝒖𝟔 is the previous term.

General nth term: 𝒖𝒏 ‘u sub n’i.e.) If 𝒖𝟕 = 25, then 25 is the value of the 7th term in the sequence

(7, 25) is the ordered pair graphed on a coordinate plane

Recursive formula: a starting value & a recursive rule for generating a sequence.

Arithmetic Formula Geometric Formula

ቊ𝑢1 = 1𝑠𝑡 𝑡𝑒𝑟𝑚𝑢𝑛 = 𝑢𝑛−1 + 𝒅

s.t. n ≥ 2 ቊ𝑢1 = 1𝑠𝑡 𝑡𝑒𝑟𝑚𝑢𝑛 = 𝒓 ∙ 𝑢𝑛−1

s.t. n ≥ 2

Recursive rule: defines the nth term of a sequence in relation to the previous term.

𝒖𝒏 𝒖𝒏−𝟏

Recursive Formula & Rule - EXAMPLE

(x-value) n

(y-value) u

Is this recursive sequence ARITHMETIC or GEOMETRIC ? Graph if unsure… (linear) (exponential)

r = common ratio*

Recursive Sequence: requires at least one term to be specified AND each term depends on the term before it.

*Warning: the r-value in the geometric sequence is equal to the b-value in the EXPLICIT equation y = a·𝒃𝒙

d = common differenceEXPLICIT equivalent equation: y = mx + b

#1.) Identify each sequence as arithmetic, geometric or neither.If arithmetic, identify the d-value. If geometric identify the r-value.

a. 14, 7, 3.5, 1.75,…

b. 47, 41, 35, 29,…

c. 1, 1, 2, 3, 5, 8,…

d. The non-horizontal cards

#1.) Identify each sequence as arithmetic, geometric or neither.If arithmetic, identify the d-value. If geometric identify the r-value.

a.) 14, 7, 3.5, 1.75,…

Geometric r = ½b.) 47, 41, 35, 29,…

Arithmetic d = -6c.) 1, 1, 2, 3, 5, 8,…

Fibonacci Sequence

𝒂𝟏 = 1

𝒂𝟐 = 1

𝒂𝒏 = 𝒂𝒏−𝟏+ 𝒂𝒏−𝟐 (n ≥ 3)

d.) The non-horizontal cards

Card house: {2, 4, 6, 8} finite set

Arithmetic d = 2

SOLUTION:

ቊ𝑢1 = 47𝑢𝑛 = 𝑢𝑛−1 − 𝟔

s.t. n ≥ 2

ቊ𝑢1 = 14𝑢𝑛 = 𝟎. 𝟓 ∙ 𝑢𝑛−1

s.t. n ≥ 2

ቊ𝑢1 = 2𝑢𝑛 = 𝑢𝑛−1 + 𝟐

s.t. 𝟐 ≤ 𝐧 ≤ 𝟒

Exercises: p.36

#1.) Match each description of a sequence to its recursive formula.#2.) Write the first 4 terms, tell if it is arithmetic or geometric

& identify either the d-value or r-value.a.) The first term is -18. Keep adding 4.3.

___ ___ ___ ___ _______ _____b.) Start with 47. Keep subtracting 3.

___ ___ ___ ___ _______ _____ c.) Start with 20. Keep adding 6.

___ ___ ___ ___ _______ _____d.) The first term is 32. Keep multiplying by 1.5.

___ ___ ___ ___ _______ ________

___

__

___

SOLUTIONS: Exercises: p.36 #(1, 2, 5, 6)

#1.) Match each description of a sequence to its recursive formula.#2.) Write the first 4 terms, tell if it is arithmetic or geometric & give d-value or r-value.

a.) The first term is -18. Keep adding 4.3. -18, -13.7, -9.4, -5.1 Arithmetic d = 4.3

b.) Start with 47. Keep subtracting 3. 47, 44, 41, 38 Arithmetic d= -3

c.) Start with 20. Keep adding 6. 20, 26, 32, 38 Arithmetic d = 6

d.) The first term is 32. Keep multiplying by 1.5. 32, 48, 72, 108 Geometric r = 1.5

C

B

D

A

Exercises: p.36 #(1, 2, 5 & 6)

#5.) Write a recursive formula for each sequence. Then find the indicated term.

a.) 2, 6, 10, 14, . . . Find the 15th term.

b.) 0.4, 0.04, 0.004, 0.0004, … Find the 10th term.

c.) -2, -8, -14, -20, -26, . . . Find the 11th term.

d.) -6.24, -4.03, -1.82, 0.39, … Find the 9th term.

SOLUTIONS: Exercises: p.36 #(1, 2, 5 & 6)

#5.) Write a recursive formula for each sequence. Then find the indicated term.

a.) 2, 6, 10, 14, . . . Find the 15th term. ቊ𝒂𝟏 = 𝟐

𝒂𝒏 = 𝒂𝒏−𝟏 + 𝟒where n ≥ 2

𝒂𝟏𝟓 = 58

b.) 0.4, 0.04, 0.004, 0.0004, …Find the 10th term. ቊ𝒂𝟏 = 𝟎. 𝟒

𝒂𝒏 = 𝟎. 𝟏𝒂𝒏−𝟏where n ≥ 2

𝒂𝟏𝟎 = 0.0000000004

c.) -2, -8, -14, -20, -26, . . . Find the 11th term. ቊ𝒂𝟏 = −𝟐

𝒂𝒏 = 𝒂𝒏−𝟏 − 𝟔where n ≥ 2

𝒂𝟏𝟏 = -62

d.) -6.24, -4.03, -1.82, 0.39, … Find the 9th term.

ቊ𝒂𝟏 = −𝟔. 𝟐𝟒

𝒂𝒏 = 𝒂𝒏−𝟏 + 𝟐. 𝟐𝟏where n ≥ 2

𝒂𝟗 = 11.44

Exercises: p.36 #(1, 2, 5 & 6)

#5.) Write a recursive formula for the sequence graphed. Find the 8th term.

Solutions: Exercises: p.36 #(1, 2, 5 & 6)

#5.) Write a recursive formula for the sequence graphed. Find the 8th term.

ቊ𝒖𝟏 = 𝟒𝒖𝒏 = 𝒖𝒏−𝟏 + 𝟓

𝒔. 𝒕. 𝒏 ≥ 𝟐

Let n = 5𝑢5 = 𝒖𝟓−𝟏 + 5𝑢5 = 𝒖𝟒 + 5𝑢5 = 𝟏𝟗 + 5𝑢5 = 24

{4, 9, 14, 19, ___ , ___ , ___ , ___ , …}

{4, 9, 14, 19, 24, 29, 34, 39, …}

Let n = 8𝑢8 = 𝒖𝟖−𝟏 + 5𝑢8 = 𝒖𝟕 + 5𝑢8 = 𝟑𝟒 + 5𝑢8 = 39

…

𝒖𝟖 = 𝟑𝟗

Math Studies SL Formulas