Transcript Arithmetic Sequences and Series

Arithmetic
Sequences
and Series
An Arithmetic Sequence is
defined as a sequence in
which there is a common
difference between
consecutive terms.
Which of the following sequences are
arithmetic? Identify the common
difference.
3,  1, 1, 3, 5, 7, 9, . . .
YES
15.5, 14, 12.5, 11, 9.5, 8, . . .
d 2
YES d  1.5
84, 80, 74, 66, 56, 44, . . . NO
8, 6,  4, 2, 0, . . .
NO
50,  44,  38,  32,  26, . . . YES d  6
26,  21,  16,  11,  6, . . .
The general form of an ARITHMETIC sequence.
First Term:
a1
Second Term:
a2  a1  d
Third Term:
a3  a1  2d
Fourth Term:
a4  a1  3d
Fifth Term:
a5  a1  4d
nth Term:
an  a1   n 1 d
Formula for the nth term of an ARITHMETIC
sequence.
an  a1   n 1 d
an  The nth term
a1  The 1st term
n  The term number
d  The common difference
Given: 79, 75, 71, 67, 63, . . .
Find: a32
IDENTIFY
a1  79
SOLVE
an  a1   n  1 d
d  4
a32  79   32  1 4 
n  32
a32  45
Given: 79, 75, 71, 67, 63, . . .
Find: What term number is -169?
IDENTIFY
a1  79
d  4
an  169
SOLVE
an  a1   n  1 d
169  79   n  1 4 
n  63
Given:
a10  3.25
Find: a1
a12  4.25
What’s the real question?
IDENTIFY
a1  3.25
a3  4.25
n3
The Difference
SOLVE
an  a1   n  1 d
4.25  3.25   3  1 d
d  0.5
Given:
a10  3.25
Find: a1
a12  4.25
IDENTIFY
a10  3.25
SOLVE
an  a1   n  1 d
d  0.5
3.25  a1  10  1 0.5
n  10
a1  1.25
50
  73  2 p   71 69  67  . . .  25  27
p 1
71  69  67  . . .   25   27 
 27   25  . . .  67  69  71
44  44  44  . . .  44  44  44
50 Terms

50  71   27  
2
 1100
71 + (-27) Each sum
is the same.
a1   a1  d    a1  2d   . . .   a1   n 1 d 
 a   n 1 d   . . .   a
1
1
 2d    a1  d   a1
 a  a   n 1 d     a  a   n 1 d    . . .   a  a   n 1 d  
1
s
1
1
n  a1  an 

2
1
1
1
S  Sum

 n  Number of Terms


a1  First Term


an  Last Term

S
Find the sum of the terms of this
arithmetic series. 35
 29  3k 

n  a1  an 
k 1
2
n  35
a1  26
a35  76
35  26  76 
S
2
S  875
Find the sum of the terms of this arithmetic
series.
151  147  143  139  . . .   5
n  a1  an 
S
2
n  40
a1  151
a40  5
What term is -5?
an  a1   n  1 d
5  151   n  1 4 
n  40
40 151  5 
S
2
S  2920
Substitute an  a1   n 1 d
n  a1  an 
S
2
S
S
n  a1  a1   n  1 d 
2
n  2a1   n  1 d 
2
 n  # of Terms

 a1  1st Term
 d  Difference

36
Find the sum of this series
  2.25  0.75 j 
j 0
 2.25  3  3.73  4.5  . . .
S
n  2a1   n  1 d 
n  37
2
37  2  2.25   37  1 0.75 
a1  2.25
S
d  0.75
S  582.75
2
35
  45  5i 
n  a1  an 
S
2
n  35 a1  40 an  130
35  40  130 
S
2
S  1575
i 1
S
n  2a1   n  1 d 
2
n  35 a1  40 d  5
S
35  2  40    35  1 3
S  1575
2