Sequences
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.
The notation of an arithmetic sequence is
For the SAT, you should be able to do three things with an arithmetic sequence:
  1. Find the constant interval between terms.
  2. Find any term in the sequence.
  3. 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:
In an arithmetic sequence, if a4 = 4 and a7 = 10, find d.
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:
  1. Find r.
  2. Find the nth term.
  3. 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:
If the first two terms of a sequence are 1 and 2, and all the following terms in the sequence are produced by subtracting from the previous term the term before that, then what is the fiftieth term in the sequence?
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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