Finding Patterns in SAT Math, Part II

In our last post, we went over some examples of the kinds of pattern questions that pop up on the SAT. But pattern questions come in all shapes and sizes, and in this post we’re going to look at a couple of more tricky ones.

The first type that we’re going to look at is a question that asks about nth terms of the sequence, where n is extremely high (these are also known as annoying sequences). Remember that if the SAT asks you about the 75^th or the 103^rd term of a sequence, there is no way they expect you to calculate all of the terms up to this term. There will always be some kind of shortcut.

Here’s an example:

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0,…

The sequence above consists of only 0s and 1s. The first 0 is followed by one 1, the second 0 is followed by two 1s, the third 0 is followed by three 1s, and so on. What is the total number of 1s between the 78^th and 82^nd 0 in this sequence?

(A)   278
(B)   298
(C)   308
(D)   318
(E)    328

Are we supposed to calculate this sequence up until the 82^nd 0 in the sequence? Of course not. There is a pattern and we can figure out the answer without calculating out the whole sequence.

First, let’s make sure we understand the question. We need to calculate the total number of 1s that fall between the 78^th and the 82^nd 0. This means we need to figure out the number of the 1s that follow the 78^th 0, the 79^th 0, the 80^th 0,  and the 81^st 0. (Note: we are not calculating the number of 1s after the 82^nd 0.)

Let's look at the information given in the question:

The pattern is pretty simple: the nth 0 is followed by n 1s. What does this mean for the 78^th, 79^th, 80^th, and 81^st 0s?

Add them up and you've got 78 + 79 + 80 + 81, or 318 0s. The answer is (D).

Sometimes a pattern question won’t tell you what the pattern is. You'll need to determine the pattern yourself, but the question will always give you the necessary information. Take this example:

4, 8, 14, 23, …

The first term in the above sequence is 4. Every term after the first is determined by multiplying the preceding term by q and then adding p. What is the value of q?

(A)   1
(B)   1.25
(C)   1.5
(D)   1.75
(E)    2

It’s possible to find the answer by guess-and-check, but that might take a while. You have the possible values of q, but not p. You would need to figure out the possible values of p yourself.

Don’t overlook the fact that the question gives you the first values of the sequence. Because you know the general pattern (multiply by q, add p), you can formulate a few equations. You have two variables, so you will need two equations. How about the solving the following equations?

2^nd Term = (1^st Term * q) + p

3^rd Term = (2^nd Term * q) + p

You know the 1^st, 2^nd, and 3^rd terms, so you can calculate for q and p.

8 = 4q + p (Equation I)

14 = 8q + p (Equation II)

You're left with your standard system of equations. Rearrange Equation I:

p = 8 – 4q

Plug this new value of p into Equation II:

14 = 8q + (8 – 4q)

Combine like terms:

14 = 4q + 8

6 = 4q

Solve for q:

q = 1.5

The answer is (C). You don’t need to solve for p because the question doesn’t ask for it (p = 2, if you’re curious).

Want some more practice? Try this quiz.

Got a math problem you can’t solve? Drop it in the comments or email it to testpreptutor@sparknotes.com.

Related Post: Finding Patterns in SAT Math