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