Understanding 8.1…
Use sigma notation to write the sum of
1 2 6 24 120 720
  


2 4 8 16 32 64
6
k ! 73


k
4
k 1 2
Arithmetic Sequences
8.2
JMerrill, 2007
Revised 2008
Sequences
A Sequence:



Usually defined to be a function
Domain is the set of positive integers
Arithmetic sequence graphs are linear
(usually)
Sequences
SEQUENCE - a set of numbers, called
terms, arranged in a particular order.
There are two basic types:


Arithmetic
Geometric
This unit deals with arithmetic sequences
Arithmetic Sequences
ARITHMETIC - the difference of any two consecutive
terms is constant.
In order to find the difference, you MUST pick one term
and subtract the preceding term
You MUST check more than 1 pair of terms!
2,6,10,14,18………
difference = 4
17,10,3,-4,-11,-18…….
difference = -7
a, a+d, a+2d, a+3d………….
difference = d
Are you Ready???
The difference of 8, 3, -2, -7…
is 5
True or False?
The difference of 23, 17, 11, 5, is -6
True or False?
Formulas for the nth term of a
Sequence
Arithmetic:
an = a1 + (n-1)d
To get the nth term, start with the 1st term
and add the difference (n-1) times
n = THE TERM NUMBER
Example
Find a formula for an and sketch the graph
for the sequence 1, 4, 7, 10...
Arithmetic or Geometric?
d=? 3
an = a1 + (n-1)d
an = 1 + (n-1)3
an = 1 + 3n-3
an = -2 + 3n
n = THE TERM NUMBER
Example
Find the given term of the arithmetic
sequence if a1 = 15, a2 =21, find a20
d= 6
an = a1 + (n-1)d
a20 = 15 + (19)6
a20 = 15 + 114
a20 = 129
You Do
Find a formula for the nth term of
an arithmetic sequence whose
common difference is 3 and
whose first term is 2
an = 3n - 1
Last Example
The 4th term of an arithmetic sequence is 20,
and the 13th term is 65. Write the first 3 terms of
the sequence?
1st use the equation:
an = a1 + (n-1)d
a4 = a1 + (4-1)d
a13 = a1 + (13-1)d
20 = a1 + 3d
65 = a1 + 12d
20 – 3d = a1
65 – 12d = a1
20 – 3d = 65 – 12d; d = 5
Last Example
Knowing that d = 5 and the 4th term is 20, we
can subtract 5 each time and know that the
sequence is 5, 10, 15, 20…
If we had been asked to find the equation (and
we couldn’t figure out that the 1st term was 5)…
20 = a1 + 3d
20 = a1 + 3(5)
5 = a1
So, an = 5 + (n-1)5 and an = 5n
Understanding Problem
Write the 1st 5 terms of the sequence. If
the sequence is arithmetic, find the
common difference.
1
an 
n 1
1 1 1 1 1
, , , ,
2 3 4 5 6
not arithmetic
Sum of a Finite Arithmetic Series
The sum of the 1st n terms of an arithmetic
series is
n (a1  an )
Sn 
2
n
The book uses Sn   a1  an 
2
You can see that you need the first term and the nth term.
Example
Find the sum of the 1st 25 terms of the
arithmetic series 11 + 14 + 17 + 20 + …
Step 1: Find the 25th term:
an  a1  (n  1)d
a25  11  (25  1)3  83
S25
25(11  83)

 1175
2
Example
Find the sum of the cubes of the first
twenty positive integers.
So, we want S20 = 13 + 23 + 33 + …+ 203
a1 = 1
a20 = 203 = 8000
n (a1  an )
Sn 
2
20(1  8000)
S20 
2
S20  80, 010
You Do
Find the 150th partial sum of the arithmetic
sequence 5, 16, 27, 38, 49
Can you do it?
123,675
Last Example
An auditorium has 20 rows of seats.
There are 20 seats in the 1st row, 21 seats
in the 2nd row, 22 seats in the 3rd row, and
so on. How many seats are there in all 20
rows?
a1 = 20
a2 = 21
a3 = 22
d=1
an = a1 + (n-1)d
a20 = 20 + 19
a20 = 39
n(20  39)
Sn 
2
 590
Last Problem
Find the partial sum of the following
problem WITHOUT a calculator (use
formula)
250
 (1000  n)
n 1
218,625