Arithmetic Sequences and Series 2015 2 days Digital Lesson.

Arithmetic Sequences and Series

20152 days

Digital Lesson

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

3 2

3 1

n

n

3 3 1 3 2n n n

1/26/2015 Precalculus HWQ:

Simplify the factorial expression:

!!


Precalculus Warm-up

Write an expression for the apparent nth term of the sequence:

1 3 7 15 311 ,1 ,1 ,1 ,1 ,...

2 4 8 16 32

1

2 11

2

2 1

2

n

n n

n

n

a

or


An infinite sequence is a function whose domain is the set of positive integers.

a1, a2, a3, a4, . . . , an, . . .

The first three terms of the sequence an = 4n – 7 are

a1 = 4(1) – 7 = – 3

a2 = 4(2) – 7 = 1

a3 = 4(3) – 7 = 5.

finite sequence

terms


A sequence is arithmetic if the differences between consecutive terms are the same.

4, 9, 14, 19, 24, . . .

9 – 4 = 5

14 – 9 = 5

19 – 14 = 5

24 – 19 = 5

arithmetic sequence

The common difference, d, is 5.


Example: Find the first five terms of the sequence and determine if it is arithmetic.

an = 1 + (n – 1)4

This is an arithmetic sequence.

d = 4

a1 = 1 + (1 – 1)4 = 1 + 0 = 1

a2 = 1 + (2 – 1)4 = 1 + 4 = 5

a3 = 1 + (3 – 1)4 = 1 + 8 = 9

a4 = 1 + (4 – 1)4 = 1 + 12 = 13

a5 = 1 + (5 – 1)4 = 1 + 16 = 17


Determine whether or not each sequence is arithmetic.

a) -12, -7, -2, 3, 8, . . .

b) ln1, ln2, ln3, ln4, ln5, . . .

1 2 4 8 16c) , , , , ,...

3 3 3 3 3


The nth term of an arithmetic sequence has the form an = a1 + (n – 1)d

or the alternate form: an = dn + c

where d is the common difference and c = a1 – d.

2, 8, 14, 20, 26, . . . .

d = 8 – 2 = 6

a1 = 2

c = 2 – 6 = – 4

The nth term is 6n – 4.


a1 – d =

Example: Find the formula for the nth term of an arithmetic sequence whose common difference is 4 and whose first term is 15. Find the first five terms of the sequence.

an = dn + c

= 4n + 11

15,

d = 4

a1 = 15 19, 23, 27, 31.

The first five terms are

15 – 4 = 11


Graphing Utility: Find the first 5 terms of the arithmetic sequence an = 4n + 11.

List Menu:

variable beginning value

end value

• Example: Find the formula for the nth term of an arithmetic sequence whose 4th term is 18 and whose 13th term is 63. Find the 20th term of the sequence.


18 9 63d

9 45d

5d

5 2na n

20 98a


You Try: Find the formula for the nth term of an arithmetic sequence whose 10th term is 32 and whose 16th term is 50. Find the 30th term of the sequence.

3 2na n 30 92a


Try another: Find the formula for the nth term of an arithmetic sequence whose 5th term is 190 and whose 10th term is 115. Find the 15th term of the sequence.

15 265na n 15 40a


Try another: Find the formula for the nth term of an arithmetic sequence whose 10th term is -330 and whose 20th term is -450. Find the 52nd term of the sequence.


Ex: Find the sum:56

1i

i

Homework Day 1

• Pg. 573 1-41 odds only


Arithmetic Sequences and Series Day 2

2015

Digital Lesson


1/27/2015 Precalculus Warm-up :

Find a formula for the arithmetic sequence where 5 1519 89a and a

7 16na n


Precalculus Warm-up:

Simplify the factorial expression:

2 3 !

2 2 !

n

n

1

2 2 2 1 2 2 1 2 2n n n n n


The sum of the first n terms of a sequence is represented by summation notation.

1 2 3 41

n

i ni

a a a a a a

index of summation

upper limit of summation

lower limit of summation

5

1

1i

n

(11) (1 2) (1 3) (1 4) (15)

2 3 4 5 6

20


Consider the infinite sequence a1, a2, a3, . . ., ai, . . ..

1. The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence.

1

n

ii

a

a1 + a2 + a3 + . . . + an

2. The sum of all the terms of the infinite sequence is called an infinite series.

1i

i

a

a1 + a2 + a3 + . . . + ai + . . .


Example:

Find the sum: 10

1

5n


The sum of a finite arithmetic sequence with n terms is given by

1( ).2n nnS a a

5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 = ?

( )501 2755 )0 5(552nS

n = 10

a1 = 5 a10 = 50


The sum of the first n terms of an infinite sequence is called the nth partial sum.

1( )2n nnS a a

( )190 25(184) 4602

50 6 0nS

an = dn + c = 4n – 10

Example: Find the 50th partial sum of the arithmetic sequence – 6, – 2, 2, 6, . . .

a50 = 4(50) – 10 = 190


100

1

2i

n

Example: Find the partial sum.

2( ) 2( ) 2( ) 2( )1 2 3 100 2 4 6 200

a1 a100

100 ( )2

20 01 00 2S

50(202) 10,100


In an arithmetic sequence, the 20th term is 116

and the 24th term is 140.

Find the sum of the first 50 terms.

6 4na n

50 7450S

1 502, 296a a


In an arithmetic sequence, the 12th term is 25

and the 30th term is 97.

Find the sum of the first 40 terms.

4 23na n

40 2360S

1 4019, 137a a


Graphing Utility: Find the first 5 terms of the arithmetic sequence an = 4n + 11.

List Menu:

variable beginning value

end value

Graphing Utility: Find the sum 100

1

2 .i

n

List Menu:

lower limit

upper limit


• Example: Find the 150th partial sum of the sequence: 5, 16, 27, 38, 49, …

150 123,675S


A stadium has 20 rows of seats. There are

20 seats in row 1, 21 in row 2, 22 in row 3,

etc. How many total seats are there?

590


Find a formula to represent the sum of n positive odd integers.

21 2 1 22 2n

n nS n n n

Homework Day 2

• Pg. 573 43-81 odds only



1/27/2015 Precalculus HWQ:

Find the fourth partial sum of 1

15 .2

i

i

1 2 3 44

1

1 1 1 1 15 5 5 5 52 2 2 2 2

i

i

1 1 1 15 5 5 52 4 8 16

5 5 5 52 4 8 16

40 20 10 5 7516 16 16 16 16

Arithmetic Sequences and Series 2015 2 days Digital Lesson.

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