A sequence can be described as a series of
numbers that proceed one after another in a certain pattern. Think
of that cheerleader chant that starts with “2, 4, 6, 8!” Before
you ask who we appreciate, realize that those four numbers form
a sequence wherein the next term is two more than the previous term.
Fortunately, you don’t need an intimate knowledge of pom-pom maintenance
and usage to understand sequences.
In an arithmetic sequence, each term is greater
than the previous term by some fixed constant. In our cheerleader
cheer, the first term equals 2, and the interval is 2:
2, 4, 6, 8, . . .
The value by which the sequence increases remains the
same. This term is called the constant. In this case,
the constant—often called k—equals 2.
There are many different formulas regarding sequences,
but you’re usually better off just using your pencil and calculator
to solve most SAT sequence items. Write out the sequence until you
find what you need. It may seem low-tech, but it brings results,
and that’s all that really counts.
Geometric Sequence and Exponential Growth
Arithmetic sequences have a constant that is added to
each consecutive term. Geometric sequences have a number
that is multiplied to each consecutive term. In
geometric sequences, this constant term is called the common
Let’s take our cheerleader cheer and make it
a geometric sequence. If the first term is 2 and the common ratio
is 2, then the cheer starts off the same, because 2 + 2 = 4 and
(2)(2) = 4. After that, things go upward fast.
2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, . . .
A geometric sequence can also include negative numbers.
When the common ratio is a negative number (for example, –5), the
sequence contains both negative and positive numbers, as in the
1, –5, 25, –125, . . .
Other, Less Pleasant Sequences
Arithmetic and geometric sequences have terms
marching in lockstep, with each term affected in the same way. This
is not true of every sequence you might encounter on the SAT. Some
have no constant difference between the terms, or the difference
between the terms keeps changing.
Here is an example of one such sequence:
2, 3, 6, 18, 108, 1944, . . .
If you cannot immediately determine whether a sequence
is arithmetic or geometric, try to find the relationship between
numbers. This takes some guesswork, and there’s no magic feather
to help guide you. The SAT tests your ability to go the extra mile
by using logic. In the above sequence, each term is the product
of the two previous terms:
23 = 6, 36
= 18, 618 = 108, 18108
= 1944, . . .
You can expect to find unusual, hard-to-determine sequences
hanging out in the latter part of a Math section. On easier items,
a sequence most likely is either arithmetic or geometric.