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Sequences
A sequence is a series of numbers that proceed one after
another according to some pattern. Here’s one:
1, 2, 4, 8, 16,...
Each number in this sequence doubles the previous number.
Once you know the pattern, you can come up with the number after
16, which is 32, the number after that, which is 64, and, if you
felt like it, you could keep calculating numbers in the sequence
for the rest of your life.
The SAT tests you on three specific types of sequences:
arithmetic, geometric, and annoying.
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.
NOTE: This is the one time in the English language when the phrase “same difference” makes sense.
An example is 1, 4, 7, 10, 13, ..., where 3 is the constant increment between values.
NOTE: This is the one time in the English language when the phrase “same difference” makes sense.
An example is 1, 4, 7, 10, 13, ..., where 3 is the constant increment between values.
The notation of an arithmetic sequence is
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For the SAT, you should be able to do three things with
an arithmetic sequence:
- Find the constant interval between terms.
- Find any term in the sequence.
- Calculate the sum of the first n terms.
Finding the Constant Interval (a.k.a., Finding d)
To find the constant interval, d, just
subtract one term in an arithmetic sequence from the next. For the
arithmetic sequence an =
1, 4, 7, 10, 13, ..., 
.
.Okay, now here’s a slightly more complicated
form of this same d-finding question:
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This question gives you the fourth and seventh terms of
an arithmetic sequence:
Since in arithmetic sequences d is constant
between every term, you know that a4 + d = a5, a5 + d = a6,
and a6 + d =
10. In other words, the difference between the seventh term, 10,
and the fourth term, 4, is 3d.
Stated as an equation,
Now solve it.
Finding Any Term in the Sequence (a.k.a., Finding
the nth Term)
Finding the nth term is a piece of cake
when you have a formula. And we have a formula:
where an is
the nth term of the sequence and d is
the difference between consecutive terms.
So, to find the 55th term in the arithmetic sequence an =
1, 4, 7, 10, 13, ..., plug the values of a1 =
1, n = 55, and d = 3 into the
formula: 
.
.Finding the Sum of the First n Terms
Finding the sum of the first n terms
is also cake-like in its simplicity when you have a formula. And
we do:
Using the same example, the sum of the first 55 terms
would be
Geometric Sequences and Exponential Growth
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 differ 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
As with arithmetic sequences, you should be able to perform
three tasks on geometric sequences for the SAT:
- Find r.
- Find the nth 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 
.
.Finding the nth Term
Want to find the nth term of a geometric
sequence? How about a formula to help you on your quest?
Here’s the formula in action. The 11th term of the sequence
3, 6, 12, 24, ... is
Finding the Sum of the First n Terms
One final formula. To find the sum of the first n terms
of a geometric sequence, use this one:
So the sum of the first 10 terms of the same sequence
is
Geometric Sequences and Negative Numbers
A geometric sequence is formed when each term is multiplied
by some standard number to get the next phrase. So far we’ve only
dealt with circumstances where that standard number was positive.
But it can also be negative. Take a sequence that starts with the
number 1 and multiplies each term by –2: 1,
–2, 4, –8, 16, –32,... See the pattern? Whenever r is
negative in a geometric sequence, the terms will alternate between
positive and negative.
Annoying Sequences
Annoying sequences is a technical math term that we just
made up. We made it up for one reason: These sequences annoy us,
and we think they’ll annoy you. Notice, though, that we didn’t name
them devastating sequences, or even difficult sequences. That’s
because they’re neither difficult nor devastating. Just annoying.
In annoying sequences, the SAT makes up the rules. For
instance, in the annoying sequence 1, 2, 3, 5, 8, 13, ...,
there isn’t any standard change between each term, but there is
a pattern: after the first two terms, each term is equal to the
sum of the previous two terms.
Annoying sequences most commonly show up in problems that
ask you to find terms at absurdly high values of n.
Here’s an annoying sequence word problem:
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The 50th term? How are you, with your busy life and no
magic formula, supposed to write out the sequence until you get
to the 50th term? Looks like you’ll end up going to college in Siberia.
While Siberia is nice for one
day each year in July, you don’t have to worry. Whenever the SAT
asks a question involving an insanely high term in a sequence, there’s always
a trick to finding it quickly. When the term is in an annoying sequence,
the trick is usually a repeating pattern that will make the answer
easy to find. So start writing out the sequence and look for the
pattern. Once you see it, strike:
1, 2, 1, –1, –2, –1, 1, 2, 1, –1,...
Do you see the pattern? After six terms, this sequence
starts to repeat itself: 1, 2, 1, –1, –2, –1 and then
it starts over. So if the sequence repeats every six terms, then every
sixth term will be a –1: the sixth term, the 12th
term, all the way up to the 48th term. And if you know
that the 48th term is a –1 and that the
sequence starts over on the 49th, then you know that
the 49th term will be a 1 and the 50th
term will be a 2.
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