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Unit 1 Sequences (Ch. 1 ) & Series (Ch. 9)
F 502. Find the next term in a sequence described recursively
F603. Find a recursive expression for the general term in a sequence described
recursively
AF 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
DO
NOW:
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.
Ch. 1.1 Recursive Sequences
Learning Intentions:
 Discover recursive formulas for sequences.
 Define, explore & use arithmetic & geometric sequences.
 Use recursively defined sequences to model real-life situations.
Vocabulary:
Recursion: applying a procedure repeatedly, starting with a number or
geometric figure, to produce a sequence of numbers or figures.
~ Each term or stage builds on the previous term or stage.
Sequence: an ordered list of numbers.
Term: each number in the sequence.
1st term / starting value: 𝒖𝟏 or 𝒂𝟏 ‘u or a sub 1’ (𝒂𝟐 = 2nd term…)
Previous term: 𝒖𝒏−𝟏 or 𝒂𝒏−𝟏
General nth term: 𝒖𝒏 or 𝒂𝒏 ‘u or a sub n’
Recursive formula: a starting value & a recursive rule for generating a sequence.
Recursive rule: defines the nth term of a sequence in relation to the previous term.
Recursive Formula & Rule - EXAMPLE
(x-value) n
(y-value) u
Is this recursive sequence ARITHMETIC or GEOMETRIC ? Graph if unsure…
(linear)
(exponential)
#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
SOLUTION:
#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 = ½
b.) 47, 41, 35, 29,…
Arithmetic d = -6
c.) 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, …
(ignoring the cards laid parallel to the floor because
this doesn’t have to be a consistent amount)
Arithmetic d = 2
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
c.) Start with 20. Keep adding 6.
20, 26, 32, 38
d.) The first term is 32. Keep multiplying by 1.5.
C
B
D
A
32, 48, 72, 108
Arithmetic d= -3
Arithmetic d = 6
Geometric r = 1.5
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.
𝒂𝟏 = 𝟐
where n ≥ 2
𝒂𝒏 = 𝒂𝒏−𝟏 + 𝟒
𝒂𝟏𝟓 = 58
b.) 0.4, 0.04, 0.004, 0.0004, …Find the 10th term.
𝒂𝟏 = 𝟎. 𝟒
where n ≥ 2
𝒂𝒏 = 𝟎. 𝟏𝒂𝒏−𝟏
𝒂𝟏𝟎 = 0.0000000004
c.) -2, -8, -14, -20, -26, . . . Find the 11th term.
𝒂𝟏 = −𝟐
where n ≥ 2
𝒂𝒏 = 𝒂𝒏−𝟏 − 𝟔
𝒂𝟏𝟏 = -62
d.) -6.24, -4.03, -1.82, 0.39, … Find the 9th term.
𝒂𝟏 = −𝟔. 𝟐𝟒
where n ≥ 2
𝒂𝒏 = 𝒂𝒏−𝟏 + 𝟐. 𝟐𝟏
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.
𝒂𝟏 = 𝟒
𝒘𝒉𝒆𝒓𝒆 𝒏 ≥ 𝟐
𝒂𝒏 = 𝒂𝒏−𝟏 + 𝟓
𝒂𝟖 = 𝟑𝟗