Arithmetic Sequences
and Series

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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  a2 - a1 is 3.
15, 10, 5, 0, -5, -10, ... d = -5 add -5 to each term to arrive at the next term,
or...the difference  a2 - a1 is -5.
add -1/2 to each term to arrive at the next term,
or....the difference a2 - a1 is -1/2.


Formulas used with arithmetic sequences and arithmetic series:

To find any term
of an
arithmetic sequence:

where a1 is the first term of the sequence,
d is the common difference, n is the number of the term to find.

Note:  a1 is often simply referred to as a.

To find the sum of a certain number of terms of an arithmetic sequence:

where Sn is the sum of n terms (nth partial sum),
 a1 is the first term,  an is the nth 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
                         an = 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 10th term of the sequence
                          3, 5, 7, 9, ...
3. n = 10;  a1 = 3, d = 2

 
The tenth term is 21.

4.  Find a7 for an arithmetic sequence where
                  a1 = 3x and d = -x.
4.  n = 7;  a1 = 3x, d = -x

5.  Find  t15 for an arithmetic sequence where
          t3 = -4 + 5i  and  t6 = -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 t3 to t15 is 13 terms.
t15 = -4 + 5i + (13-1)(-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 25th term of the sequence
                       -7, -4, -1, 2, ...
7.  n = 25;  a1 = -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;  a1 = 2, d = 2
We are missing a12, 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, a1,  the last term, an, 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.   a1 = 7, an = 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,  a1 = 60,  d = 8 and we need a20 for the sum.

Now, use the sum formula:

There are 2720 seats.

 

Check out how to use your TI-83+/84+ graphing calculator with sequences and series. Click here.