Series
A series is a sequence of numbers that proceed one after
another according to some pattern. Usually the SAT will give you
a few numbers in a series and ask you to specify what number should
come next. For example,
–1, 2, –4, 8, –16
is a series. Can you figure out which number should come
after the –16? Well, in this series, each number is
multiplied by –2 to yield the next number. Therefore, 32 is
the number in the series after –16. These types of
questions ask you to be able to recognize patterns and then apply
them. Learning to recognize the patterns is key. When you look at
a pattern, try to think whether it is changing by addition or subtraction,
multiplication or division, or by exponents. There isn’t one tried-and-true
way to find a pattern. Just think critically, and use your intuition
and trial and error.
Series Problems that Seem Harder than They Are
Sometimes the SAT might show you a series and ask you
to identify the 50th number in the series or to calculate
the sum of the first 24 numbers in the series. These
questions seem difficult and time-consuming, so many students skip
them. Other students write out the series and do the math, which
does take a bit of time. Whenever you see such a question, you should
assume that there is some shortcut to the answer. For example, on
a question that asks for the 50th term in the series,
see if the series begins to repeat itself. Take the following problem:
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The
first two numbers of a series are 1 and 2. All the numbers in the
series after that are produced by subtracting from the previous
term the term before that. What is the fiftieth term in the sequence? |
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To answer this question, start writing out the sequence
1, 2, 1, –1, –2, –1, 1, 2, 1, . . .
By this time you should see that the pattern has begun
to repeat itself: the 1st term is the same as the 7th,
the 2nd is the same as the 8th . . . Since
you know the sequence repeats, you can extrapolate into the future.
If the 1st term is the same as the 7th,
it will also be the same as the 14th, 21st, 28th, 35th, 42nd,
and 49th. This repetion means that the second term
must be equal to the 50th term, so the answer is 2.
If you were given the same question but asked to figure
out the sum of the first
35 terms, you would do basically
the same thing. Once you discovered that the sequence repeats every
seven terms, you would know that the value of the first
24 terms
is equal to
terms, since terms
1–
6,
7–
12,
13–
18,
and
19–
24 will all be identical. The sum
of the first
6 terms is:
So the sum of the first 24 terms is equal
to 0.
