


Sequences
You might see one or two sequence questions on the Math
IIC. The two types of sequences tested are arithmetic and geometric
sequences.
Arithmetic Sequences
An arithmetic sequence is an ordered list of terms in
which the difference between consecutive terms is constant. In other
words, the same value or variable is added to each term in order
to create the next term: if you subtract any two consecutive terms
of the sequence, you will get the same difference. An example is
{a_{n}} =
1, 4, 7, 10, 13, . . . , where 3 is the constant increment between
values.
The notation of an arithmetic sequence is:
where a_{n} is
the n^{th} term of the
sequence and d is the difference between consecutive
terms. For the Math IIC, you must first be able to determine that
a given sequence is an arithmetic sequence. To figure out if a sequence
is arithmetic, take two sets of consecutive terms, and subtract
the smaller term from the larger. If the difference between the
terms in the two sets is equal, you’ve got yourself an arithmetic
sequence. To figure out if the sequence {a_{n}}
= 1, 4, 7, 10, 13, . . . is arithmetic, take two sets of consecutive
terms {1, 4} and {10, 13}, and subtract the first from the second:
Since the difference is equal, you know this sequence
is arithmetic. You should be able to do three things with an arithmetic
sequence:
 Find d
 Find the nth term
 Calculate the sum of the first n terms
Finding d
To find the difference, d, between the
terms of an arithmetic sequence, just subtract one term from the
next term. The difference is d. For the arithmetic
sequence a_{n} =
1, 4, 7, 10, 13, . . . , d = 4 – 1 = 3. Here’s
a slightly more complicated form of this question:

This question gives you the fourth and seventh terms of
a sequence:
Since in arithmetic sequences d is constant
between every term, you know that d + 4 = a_{5}, a_{5} + d = a +
6, and a_{6} + d =
10. In other words, the difference between the seventh term, 10,
and the fourth term, 4, is 3d. Stated as an equation:
Solving this equation is a process of simple algebra.
Finding the nth Term
To find the n^{th} term
in an arithmetic sequence, use the following formula:
In the example above, to find the 55^{th} term,
we would have to find the value of a_{1} first.
Plug the values of a_{4} =
4, n = 4, and d = 2 into the formula a_{n} = a_{1} +
(n – 1)d to find that a_{1} equals
–2. Now find the 55^{th} term: a_{55} =
–2 + (55 – 1)2 = –2 + (54)2 = –2 + 108 = 106.
Calculating the Sum of the First n Terms
In order to find the sum of the first n terms,
simply find the value of the average term and then multiply that
average by the number of terms you are summing.
As you can see, this is simply n times
the average of the first n terms. The sum of the
first 55 terms of the above sequence would be sum = 55[(–2 + 106)
⁄ 2] = 52(55) = 2,860.
Geometric Sequences
A geometric sequence is a sequence in which the ratio
of any term and the next term is constant. Whereas in an arithmetic
sequence the difference between consecutive terms
is always constant, in a geometric sequence the quotient of
consecutive terms is always constant. The constant factor by which
the terms of a geometric function differs is called the common ratio
of the geometric sequence. The common ratio is usually represented
by the variable r. Here is an example
of a geometric sequence in which r = 3.
The general form of a geometric sequence is:
You should be able to identify a geometric sequence from
its terms, and you should be able to perform three tasks on geometric
sequences:
 Find r
 Find the n^{th} term
 Calculate the sum of the first n terms
Finding r
To find the common ratio of a geometric sequence, all
you have to do is divide one term by the preceding term.
For example, the value of r for the sequence
3, 6, 12, 24, . . . is ^{6}
/_{3} = 2.
Finding the nth Term
To find the n^{th} term
of a geometric sequence, use the following formula:
For example, the 11^{th} term
of the sequence above is:
Finding the Sum of the First n Terms
To find the sum of the first n terms
of a geometric sequence, use the following formula:
So, the sum of the first 10 terms of the same sequence
is:
The Sum of an Infinite Geometric Sequence
If the common ratio r of a geometric
sequence is greater or equal to 1 (or less than or equal to –1),
then each term is greater than or equal to the previous term, and
the sequence does not converge. For these sequences, we cannot find
a sum.
But if –1 < r < 1, then the terms
of the sequence will converge toward zero. This convergence toward
zero means that the sum of the entire geometric sequence can be
approximated quite closely with the following formula:
where –1 < r < 1. For example:

As a first step, make sure that the sum can be calculated
by determining r. For this sequence, r =
^{1}/_{2} ,
which is between –1 and 1, so the sum of the sequence is finite.
Now, using the formula, the sum is (1 – ^{1}
/_{2} ) = 8.
