Download - AA Section 3-7/3-8
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Section 3-7 and 3-8Recursive and Explicit Formulas for Arithmetic Sequences
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Warm-up1. Find the first four terms of the recursive sequence
S1 = 10Sn = Sn−1 + 5, for int. n ≥ 2⎧⎨⎩
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Warm-up1. Find the first four terms of the recursive sequence
S1 = 10Sn = Sn−1 + 5, for int. n ≥ 2⎧⎨⎩
S1 = 10
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Warm-up1. Find the first four terms of the recursive sequence
S1 = 10Sn = Sn−1 + 5, for int. n ≥ 2⎧⎨⎩
S1 = 10 S2 = 15
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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
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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
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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, ...
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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⎧⎨⎩
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Arithmetic Sequence
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Arithmetic Sequence
A sequence with a constant difference between terms
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Arithmetic Sequence
A sequence with a constant difference between terms
What’s the domain?
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Arithmetic Sequence
A sequence with a constant difference between terms
What’s the domain?
The set of natural numbers
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Theorem
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Theorem(Recursive Formula for an Arithmetic Sequence)
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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⎧⎨⎩
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example 1Consider the sequence
a1 = 53an = an−1 − 7, for int. n ≥ 2
⎧⎨⎩
a. Describe the sequence in words
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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
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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.
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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
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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
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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
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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
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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
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QuestionWhat should this graph look like?
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QuestionWhat should this graph look like?
Domain is natural numbers
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QuestionWhat should this graph look like?
Domain is natural numbers
Will be discrete
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QuestionWhat should this graph look like?
Domain is natural numbers
Will be discrete
Will be a line (discrete)
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QuestionWhat should this graph look like?
Domain is natural numbers
Will be discrete
Will be a line (discrete)
Also known as a linear sequence
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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.
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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
⎧⎨⎩
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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*
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Example 3Find a explicit formula for the arithmetic sequence
12, 14.5, 17, 19.5, ...
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Example 3Find a explicit formula for the arithmetic sequence
12, 14.5, 17, 19.5, ...
a1 = 12
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Example 3Find a explicit formula for the arithmetic sequence
12, 14.5, 17, 19.5, ...
a1 = 12
a2 = 12 + 2.5
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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)
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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
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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)
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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
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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)
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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
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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
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Theorem
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Theorem(Explicit Formula for an Arithmetic Sequence)
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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
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Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
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Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
a1 = 27
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Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
a1 = 27 d =
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Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
a1 = 27 d = 28.5 - 27
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Example 4Find the 75th term of the arithmetic sequence 27, 28.5, 30,
31.5, ...
a1 = 27 d = = 1.528.5 - 27
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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
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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
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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
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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
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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
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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
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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?
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Homework
![Page 67: AA Section 3-7/3-8](https://reader031.vdocuments.us/reader031/viewer/2022012923/58f2e7761a28ab5a288b45ed/html5/thumbnails/67.jpg)
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