6.8 SAT Math - Sets and Sequences

Sets

Sets consist of questions dealing with Union and Intersection.  The numbers in a set are often called elements, or members of a set. 

Union is asked for by this symbol: ∪ and means all numbers within those sets. 
Intersection is asked for by this symbol: ∩ and means only those numbers shared by both sets.

-SAT Math Hint: An easy way to remember which is which: The union symbol looks like a wide 'u' and the Intersection symbol looks like a wide 'n'.

In other words, union, just like in history class, is where they all come together in a group.  Intersection, just like in a car, is only where they crash.

So as an example, if you are given

Set X: {1, 2, 3, 4, 5, 6}

Set Y: {2, 4, 6, 8, 10 }

X ∪ Y would equal {1, 2, 3, 4, 5, 6, 8, 10 } and X ∩ Ywould equal {2, 4, 6 }.

Union and intersection problems don't show up that often and generally aren't that difficult, but keep in mind that they are often testing definitions: so remember the vocab from the beginning.  Know what groups zero belongs to, for instance, or what rational numbers are.

Sequences

Sequences are another type of pattern problems.  They can vary from pretty simple questions like, "In the sequence 7, 11, 15, 19…what is the next number?" to pretty complex geometric sequences. 

The simplest type of sequence problem is the kind in which you are simply asked to follow steps to get the next number.  Know the words term (a number or expression in the list), preceding (before) and subsequent (following).  Take a look:

After the first term in a sequence, each subsequent term is 4 more than double the preceding term.  If the first term of the sequence is 1, what is the third term?

(A) 2
(B) 6
(C) 8
(D) 12
(E) 20

Just like translation problems, simply write out the sequence, starting with the number 1, and following the steps to get to the term asked for.

A more complicated problem may combine sequences and the pattern problems we talked about when we discussed remainders.  On these types of questions, carry out the sequence at least 6 or 8 terms – the pattern usually repeats every three to five terms, and you'll be able to catch it if you carry it out far enough.  Try this one:

-3, -9, 9...

The first term in the sequence shown above is -3.  Each even-numbered term is 6 less than the previous term and each odd-numbered term after the first is -1 times the previous term.  What is the 51st term in the sequence?

To solve these, first carry out your sequence so you can determine the pattern.
-SAT Math Tip: Follow the rules as written, not what it may look like to you!  You may see that the second term in the sequence is 3 times the first, but that is NOT how the sequence is being determined according to the question.  Use the steps given.

Once you carry out the sequence, you start to see a pattern emerge: -3,-9,9,3,-3,-9,9,3…  Keeping in mind that pattern problems are really remainder problems in disguise, what you need to figure out is how many terms are in the pattern – that will be the number you are dividing by.  In this case, there are four numbers in the pattern and they are asking for the 51st.  That means you need to find the remainder when 51 is divided by 4, find that number in your pattern, and choose the answer.

(A) -9
(B) -3
(C) 0
(D) 3
(E) 9

The answer is E) 9.

Arithmetic and Geometric Sequences

Even harder sequence problems will simply ask you to carry out a sequence for a while.  These look and sound simple, and in fact they are, they just take a really long time to do if you do them in the traditional manner.

There are two ways to do these types of questions:

1) You can simply follow the directions and write out the sequence until you get the number they're looking for.
2) You can memorize two sequence formulas as shortcuts.


If you are good at memorizing, know the two formulas that are about to follow and use them when you can.  If you are not good at memorizing, you may want to skip sequence questions!  It doesn't matter that the math is easy; sequence questions are often there simply to waste your time.  If you can't remember the formula that goes with them, you're most likely better off skipping these questions and only coming back if you have the time to spend.

So here are the formulas:

Arithmetic: a₁ + (n - 1)d where a=the first term in the sequence, n=the term you are trying to find the value of, and d=the difference between terms.

An arithmetic sequence is one in which each subsequent number in the sequence is simply a certain number plus or minus the previous.  Sequences like 5, 8, 11, 14 are arithmetic as are 4, -1, -6, -11.

So the question might say, "In a certain arithmetic sequence, a₁ = 1/2 and a₂ = 4.  What is the value of a₆₃?"  What this means is that the first term is 1/2, the second term is 4, and they are asking you for the 63rd term.  To use the formula, subtract the first term from the second to find the difference the sequence is based on: .  Then put these numbers into the formula:

Starting Score 600 and Above:

You may also be asked to find the sum of a certain number of terms in an arithmetic sequence.  In that case, you will need both the formula above, to find the term they're asking about, as well as another formula.

Sum of Terms in an Arithmetic Sequence:  where S_n represents the sum of the terms, n is the number of terms you are adding, a₁ is the first term in the sequence, and a_n is the final term specified in the sequence (the one you found with the earlier formula.)  So let's take a look:

If S_n is equal to the sum of all values of a_n where a₁ = 3, a₂ = 7, a₃ = 11..., what is the sum of the first 60 terms of a_n?

Use the first arithmetic sequence to find the 60th term: 3 + (60 - 1)4 = 239 and then plug that number into the Sum formula:

A geometric sequence operates in much the same way as an arithmetic, except instead of adding or subtracting to each number to get the next, you are multiplying or dividing.  What you're looking for between each two numbers, then, is a constant ratio – that is, divide one term by the term immediately in front of it.  That's the ratio that the sequence has in common.

To find a specific term in a Geometric Sequence: a₁(r^n-1)where a₁ is the first term in the sequence, r is the common ratio, and n is the term you're looking for.

Take a look at this SAT math problem:

18. If the first term in the geometric sequence above is 2, what is the eighth term in the sequence?

First, determine the common ratio by dividing one term from the one before it.  We'll use the first two terms: , so the common ratio is 2/3.

-SAT Math Hint: Make your life easier!  If it's easier for whatever reason to do the math between the third and fourth terms rather than the first and second, use those instead.  It doesn't matter which two terms you use as long as they are next to each other.

Now plug the numbers into the formula:

The eighth term of the sequence is 256/2187.