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Sequences
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.
Arithmetic Sequence
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 ratio.
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 following example:
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.
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