AA Section 3-7/3-8
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Transcript of AA Section 3-7/3-8
Section 3-7 and 3-8Recursive and Explicit Formulas for Arithmetic Sequences
Warm-up1. Find the first four terms of the recursive sequence
S1 = 10Sn = Sn−1 + 5, for int. n ≥ 2⎧⎨⎩
Warm-up1. Find the first four terms of the recursive sequence
S1 = 10Sn = Sn−1 + 5, for int. n ≥ 2⎧⎨⎩
S1 = 10
Warm-up1. Find the first four terms of the recursive sequence
S1 = 10Sn = Sn−1 + 5, for int. n ≥ 2⎧⎨⎩
S1 = 10 S2 = 15
Warm-up1. Find the first four terms of the recursive sequence
S1 = 10Sn = Sn−1 + 5, for int. n ≥ 2⎧⎨⎩
S1 = 10 S2 = 15 S3 = 20
Warm-up1. Find the first four terms of the recursive sequence
S1 = 10Sn = Sn−1 + 5, for int. n ≥ 2⎧⎨⎩
S1 = 10 S2 = 15 S3 = 20 S4 = 25
Warm-up1. Find the first four terms of the recursive sequence
S1 = 10Sn = Sn−1 + 5, for int. n ≥ 2⎧⎨⎩
S1 = 10 S2 = 15 S3 = 20 S4 = 25
2. Write a recursive formula for the sequence 150, 130, 110, 90, ...
Warm-up1. Find the first four terms of the recursive sequence
S1 = 10Sn = Sn−1 + 5, for int. n ≥ 2⎧⎨⎩
S1 = 10 S2 = 15 S3 = 20 S4 = 25
2. Write a recursive formula for the sequence 150, 130, 110, 90, ...
S1 = 150Sn = Sn−1 − 20, for int. n ≥ 2⎧⎨⎩
Arithmetic Sequence
Arithmetic Sequence
A sequence with a constant difference between terms
Arithmetic Sequence
A sequence with a constant difference between terms
What’s the domain?
Arithmetic Sequence
A sequence with a constant difference between terms
What’s the domain?
The set of natural numbers
Theorem
Theorem(Recursive Formula for an Arithmetic Sequence)
Theorem(Recursive Formula for an Arithmetic Sequence)
The sequence defined by the recursive formula
is the arithmetic sequence with first term a1 and constant difference d.
a1
an = an−1 + d, for int. n ≥ 2⎧⎨⎩
example 1Consider the sequence
a1 = 53an = an−1 − 7, for int. n ≥ 2
⎧⎨⎩
a. Describe the sequence in words
example 1Consider the sequence
a1 = 53an = an−1 − 7, for int. n ≥ 2
⎧⎨⎩
a. Describe the sequence in words
It is an arithmetic sequence with first term 53 and constant difference of -7
example 1Consider the sequence
a1 = 53an = an−1 − 7, for int. n ≥ 2
⎧⎨⎩
a. Describe the sequence in words
It is an arithmetic sequence with first term 53 and constant difference of -7
b. Find the first five terms.
example 1Consider the sequence
a1 = 53an = an−1 − 7, for int. n ≥ 2
⎧⎨⎩
a. Describe the sequence in words
It is an arithmetic sequence with first term 53 and constant difference of -7
b. Find the first five terms.a1 = 53
example 1Consider the sequence
a1 = 53an = an−1 − 7, for int. n ≥ 2
⎧⎨⎩
a. Describe the sequence in words
It is an arithmetic sequence with first term 53 and constant difference of -7
b. Find the first five terms.a1 = 53 a2 = 46
example 1Consider the sequence
a1 = 53an = an−1 − 7, for int. n ≥ 2
⎧⎨⎩
a. Describe the sequence in words
It is an arithmetic sequence with first term 53 and constant difference of -7
b. Find the first five terms.a1 = 53 a2 = 46 a3 = 39
example 1Consider the sequence
a1 = 53an = an−1 − 7, for int. n ≥ 2
⎧⎨⎩
a. Describe the sequence in words
It is an arithmetic sequence with first term 53 and constant difference of -7
b. Find the first five terms.a1 = 53 a2 = 46 a3 = 39 a4 = 32
example 1Consider the sequence
a1 = 53an = an−1 − 7, for int. n ≥ 2
⎧⎨⎩
a. Describe the sequence in words
It is an arithmetic sequence with first term 53 and constant difference of -7
b. Find the first five terms.a1 = 53 a2 = 46 a3 = 39 a4 = 32 a5 = 25
QuestionWhat should this graph look like?
QuestionWhat should this graph look like?
Domain is natural numbers
QuestionWhat should this graph look like?
Domain is natural numbers
Will be discrete
QuestionWhat should this graph look like?
Domain is natural numbers
Will be discrete
Will be a line (discrete)
QuestionWhat should this graph look like?
Domain is natural numbers
Will be discrete
Will be a line (discrete)
Also known as a linear sequence
Example 2The town of Mitarnowskia has decided to add 20,000 acre-feet of water to its reservoir capacity of 3 million acre-feet and add 20,000 acre-feet each year thereafter.
Write a recursive formula for capacity in n years.
Example 2The town of Mitarnowskia has decided to add 20,000 acre-feet of water to its reservoir capacity of 3 million acre-feet and add 20,000 acre-feet each year thereafter.
Write a recursive formula for capacity in n years.
a1 = 3,020,000an = an−1 + 20,000, for int. n ≥ 2
⎧⎨⎩
Example 2The town of Mitarnowskia has decided to add 20,000 acre-feet of water to its reservoir capacity of 3 million acre-feet and add 20,000 acre-feet each year thereafter.
Write a recursive formula for capacity in n years.
a1 = 3,020,000an = an−1 + 20,000, for int. n ≥ 2
⎧⎨⎩
*Here, a1 is 3,020,000 instead of 3,000,000 as they are going to end up having 3,020,000 acre-feet of capacity
when the situation begins*
Example 3Find a explicit formula for the arithmetic sequence
12, 14.5, 17, 19.5, ...
Example 3Find a explicit formula for the arithmetic sequence
12, 14.5, 17, 19.5, ...
a1 = 12
Example 3Find a explicit formula for the arithmetic sequence
12, 14.5, 17, 19.5, ...
a1 = 12
a2 = 12 + 2.5
Example 3Find a explicit formula for the arithmetic sequence
12, 14.5, 17, 19.5, ...
a1 = 12
a2 = 12 + 2.5 = 12 + 1(2.5)
Example 3Find a explicit formula for the arithmetic sequence
12, 14.5, 17, 19.5, ...
a1 = 12
a2 = 12 + 2.5 = 12 + 1(2.5)
a3 = 12 + 2.5 + 2.5
Example 3Find a explicit formula for the arithmetic sequence
12, 14.5, 17, 19.5, ...
a1 = 12
a2 = 12 + 2.5 = 12 + 1(2.5)
a3 = 12 + 2.5 + 2.5 = 12 + 2(2.5)
Example 3Find a explicit formula for the arithmetic sequence
12, 14.5, 17, 19.5, ...
a1 = 12
a2 = 12 + 2.5 = 12 + 1(2.5)
a3 = 12 + 2.5 + 2.5 = 12 + 2(2.5)
a4 = 12 + 2.5 + 2.5 + 2.5
Example 3Find a explicit formula for the arithmetic sequence
12, 14.5, 17, 19.5, ...
a1 = 12
a2 = 12 + 2.5 = 12 + 1(2.5)
a3 = 12 + 2.5 + 2.5 = 12 + 2(2.5)
a4 = 12 + 2.5 + 2.5 + 2.5 = 12 + 3(2.5)
Example 3Find a explicit formula for the arithmetic sequence
12, 14.5, 17, 19.5, ...
a1 = 12
a2 = 12 + 2.5 = 12 + 1(2.5)
a3 = 12 + 2.5 + 2.5 = 12 + 2(2.5)
a4 = 12 + 2.5 + 2.5 + 2.5 = 12 + 3(2.5)
an = 12 + (n - 1)2.5
Example 3Find a explicit formula for the arithmetic sequence
12, 14.5, 17, 19.5, ...
a1 = 12
a2 = 12 + 2.5 = 12 + 1(2.5)
a3 = 12 + 2.5 + 2.5 = 12 + 2(2.5)
a4 = 12 + 2.5 + 2.5 + 2.5 = 12 + 3(2.5)
an = 12 + (n - 1)2.5 for int. n ≥ 1
Theorem
Theorem(Explicit Formula for an Arithmetic Sequence)
Theorem(Explicit Formula for an Arithmetic Sequence)
The nth term of an arithmetic sequence with first term a1 and constant difference d is given by the explicit formula
an = a1 + (n − 1)d, for int. n ≥ 1
Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
a1 = 27
Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
a1 = 27 d =
Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
a1 = 27 d = 28.5 - 27
Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
a1 = 27 d = = 1.528.5 - 27
Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
a1 = 27 d = = 1.528.5 - 27
an = 27 + (n - 1)1.5
Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
a1 = 27 d = = 1.528.5 - 27
an = 27 + (n - 1)1.5a75 = 27 + (75 - 1)1.5
Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
a1 = 27 d = = 1.528.5 - 27
an = 27 + (n - 1)1.5a75 = 27 + (75 - 1)1.5
= 27 + (74)1.5
Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
a1 = 27 d = = 1.528.5 - 27
an = 27 + (n - 1)1.5a75 = 27 + (75 - 1)1.5
= 27 + (74)1.5= 27 + 111
Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
a1 = 27 d = = 1.528.5 - 27
an = 27 + (n - 1)1.5a75 = 27 + (75 - 1)1.5
= 27 + (74)1.5= 27 + 111
= 138
Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
a1 = 27 d = = 1.528.5 - 27
an = 27 + (n - 1)1.5a75 = 27 + (75 - 1)1.5
= 27 + (74)1.5= 27 + 111
= 138a75 = 138
Example 5The first row in an auditorium has 15 seats in it. Each subsequent row has 3 more seats in it than the row in
front of it. If the last row has 78 seats, how many rows are in the auditorium?
Example 5The first row in an auditorium has 15 seats in it. Each subsequent row has 3 more seats in it than the row in
front of it. If the last row has 78 seats, how many rows are in the auditorium?
a1 = 15
Example 5The first row in an auditorium has 15 seats in it. Each subsequent row has 3 more seats in it than the row in
front of it. If the last row has 78 seats, how many rows are in the auditorium?
a1 = 15 d = 3
Example 5The first row in an auditorium has 15 seats in it. Each subsequent row has 3 more seats in it than the row in
front of it. If the last row has 78 seats, how many rows are in the auditorium?
a1 = 15 d = 3an = 15 + (n - 1)3
Example 5The first row in an auditorium has 15 seats in it. Each subsequent row has 3 more seats in it than the row in
front of it. If the last row has 78 seats, how many rows are in the auditorium?
a1 = 15 d = 3an = 15 + (n - 1)378 = 15 + (n - 1)3
Example 5The first row in an auditorium has 15 seats in it. Each subsequent row has 3 more seats in it than the row in
front of it. If the last row has 78 seats, how many rows are in the auditorium?
a1 = 15 d = 3an = 15 + (n - 1)378 = 15 + (n - 1)3
-15-15
Example 5The first row in an auditorium has 15 seats in it. Each subsequent row has 3 more seats in it than the row in
front of it. If the last row has 78 seats, how many rows are in the auditorium?
a1 = 15 d = 3an = 15 + (n - 1)378 = 15 + (n - 1)3
-15-1563 = (n - 1)3
Example 5The first row in an auditorium has 15 seats in it. Each subsequent row has 3 more seats in it than the row in
front of it. If the last row has 78 seats, how many rows are in the auditorium?
a1 = 15 d = 3an = 15 + (n - 1)378 = 15 + (n - 1)3
-15-1563 = (n - 1)33 3
Example 5The first row in an auditorium has 15 seats in it. Each subsequent row has 3 more seats in it than the row in
front of it. If the last row has 78 seats, how many rows are in the auditorium?
a1 = 15 d = 3an = 15 + (n - 1)378 = 15 + (n - 1)3
-15-1563 = (n - 1)33 321 = n - 1
Example 5The first row in an auditorium has 15 seats in it. Each subsequent row has 3 more seats in it than the row in
front of it. If the last row has 78 seats, how many rows are in the auditorium?
a1 = 15 d = 3an = 15 + (n - 1)378 = 15 + (n - 1)3
-15-1563 = (n - 1)33 321 = n - 1
n = 22 rows
Homework
Homework
p. 177 # 1-6, 8-11; p. 183 #1-16, skip 8, 12
“It is impossible to defeat an ignorant man in argument.” - William G. McAdoo