A sequence is an ordered list of numbers.
The sum of the terms of a sequence is called a series.
While some sequences are simply random values,
other sequences have a definite pattern that is
used to arrive at the sequence's terms.
Two such sequences are the arithmetic and geometric
sequences. Let's investigate the arithmetic sequence.
Arithmetic Sequences 
ADD 

If a sequence of values follows a
pattern of
adding
a fixed amount from one term to the next, it is referred to as an
arithmetic sequence.
The number added to each term is constant (always the same).
The fixed amount is called the common difference,
d, referring to the fact that the difference
between two successive terms yields the constant value that was
added. To find the common difference, subtract the
first term from the second term. 

Notice the linear nature of the
scatter plot of the terms of an arithmetic sequence. The
domain consists of the counting numbers 1, 2, 3, 4, ... and the
range consists of the terms of the sequence. While
the x value increases by a constant value of one, the y
value increases by a constant value of 3 (for this graph). 
Examples:
Arithmetic
Sequence 
Common Difference, d 
1, 4, 7, 10, 13, 16, ... 
d
= 3 
add
3 to each term to arrive at the next term,
or...the difference a_{2}
 a_{1} is 3. 
15, 10, 5, 0, 5, 10, ... 
d = 5 
add
5 to each term to arrive at the next term,
or...the difference a_{2}
 a_{1} is 5. 


add
1/2 to each term to arrive
at the next term,
or....the difference a_{2}
 a_{1} is 1/2. 
Formulas used with arithmetic sequences
and arithmetic series:
To
find any term
of an arithmetic sequence:
where a_{1} is the first term of the
sequence,
d is the common difference, n is the
number of the term to find. 
Note: a_{1}
is often simply referred to as a.

To find the
sum of a certain number of
terms of an arithmetic sequence:
where S_{n} is the sum of n
terms (n^{th }partial sum),
a_{1} is the first term, a_{n} is the n^{th}
term. 

Examples:
Question 
Answer 
1.
Find the common difference for this arithmetic sequence
5, 9, 13, 17 ... 
1. The common difference, d,
can be found by subtracting the first term from the second term,
which in this problem yields 4. Checking shows that
4 is the difference between all of the entries. 
2.
Find the common difference for the arithmetic sequence whose
formula is
a_{n} = 6n + 3 
2. The formula indicates that 6 is the
value being added (with increasing multiples) as the terms
increase. A listing of the terms will also show what is
happening in the sequence (start with n = 1).
9, 15, 21, 27, 33, ...
The list shows the common difference to be 6. 
3.
Find the 10^{th} term of the sequence
3, 5, 7, 9, ... 
3. n = 10; a_{1}
= 3, d = 2
The tenth term is 21. 
4.
Find a_{7} for an
arithmetic sequence where
a_{1} = 3x and d = x. 
4. n = 7;
a_{1} = 3x, d = x

5.
Find t_{15} for
an arithmetic sequence where
t_{3} = 4
+ 5i and t_{6 }= 13 + 11i

5. Notice the change of labeling
from a to t. The letter used in labeling is
of no importance. Get a visual image of this
problem
Using the third term as the "first" term, find the common
difference from these known terms.
Now, from t_{3} to t_{15 }is 13 terms.
t_{15} = 4 + 5i + (131)(3 +2i) =
4 + 5i 36 +24i
= 40 + 29i 
6.
Find a
formula for the sequence
1, 3, 5, 7, ... 
6. A formula will
relate the subscript number of each term to the actual value of
the term.
Substituting n = 1, gives 1.
Substituting n = 2, gives 3, and so on. 
7.
Find the 25^{th}
term of the sequence
7, 4, 1, 2, ... 
7. n = 25; a_{1}
= 7, d = 3 
8. Find the sum of the
first 12 positive even
integers.

8. The
word "sum" indicates the need for the sum formula.
positive even integers: 2, 4, 6, 8, ...
n = 12; a_{1} = 2, d =
2
We are missing a_{12}, for the sum formula, so we use the
"any term" formula to
find it.
Now, let's find the sum:

9.
Insert 3 arithmetic
means between 7 and 23.
Note: An
arithmetic mean is the term between any
two terms of an arithmetic sequence. It is
simply the average (mean) of the given terms. 

9. While there are several
solution methods, we will use our arithmetic sequence formulas.
Draw a picture to better
understand the situation.
7, ____, ____, ____, 23
This set of terms will be an arithmetic sequence.
We know the first term, a_{1}, the last
term, a_{n}, but not the common difference, d.
This question makes NO mention of "sum", so avoid that formula.
Find the common difference:
Now, insert the terms using d.
7, 11, 15, 19, 23 
10.
Find the number of
terms in the sequence
7, 10, 13, ..., 55.

10. a_{1}
= 7, a_{n} = 55, d = 3. We need to
find n.
This question makes NO mention of "sum", so avoid that formula.
When solving for n, be sure
your answer is a positive integer.
There is no such thing as a fractional number of terms in a
sequence! 
11. A theater has 60
seats in the first row, 68 seats in the second row, 76 seats in
the third row, and so on in the same increasing pattern.
If the theater has 20 rows of seats, how many seats are in the
theater? 
11. The seating pattern is forming an
arithmetic sequence.
60, 68, 76, ...
We wish to find "the sum" of all of the seats.
n = 20, a_{1} = 60, d =
8 and we need a_{20
}for the sum._{
}
Now, use the sum formula:
There are 2720 seats. 

Check out how to use your
TI83+/84+ graphing calculator with
sequences and series.
Click here. 

