Sequences | Math Reasoning

Introduction

Sequence questions test the ability to detect numerical patterns in a series of numbers. The numbers given will never relate to each other randomly; there is always a pattern to discover. The CLT will ask you to supply the missing number in the pattern, which is sometimes an integer and sometimes not.

Approach Question

What is the next term in the sequence below?

-4, 8, -16, 32, ?, …

A. 64
B. -48
C. -64
D. -96

Explanation

Sequence questions on the CLT can be arranged into three categories: 1) arithmetic sequences; 2) geometric sequences; and 3) miscellaneous sequences that fall into neither of these categories. You will immediately enhance your understanding of the problem if you identify which category the sequence in question falls into.

Although this question does not label the sequence (sometimes the CLT will refer to a given sequence as “arithmetic” or “geometric”), the sequence does bear the characteristics of a common type. To discover that type, we follow the approach always necessary on sequence questions: identifying the relationship between one term and the next. Sequences always have a consistent pattern whereby one of the numbers (“terms”) relates to the one before it and the one after it. As you read the numbers from left to right, you will find that there is always some consistent way in which the numbers are changed.

There are two aspects to the pattern here. First, ignoring the signs (positive or negative), we can note that the numbers appear to be doubling each time: from $4$ , to $8$ , to $16$ , etc. Then, paying attention to positive and negative, we see an alternating pattern: the first number is negative, the second positive, the third negative, and so on.

We are called on to supply the fifth number, the one after $32$ . Doubling $32$ gives us $64$ . But that does not mean the answer is A (trap answer alert!) because that answer doesn’t take into account the alternating signs. Since every number after a positive number is negative in this sequence, and since the number before the unknown number is positive, our answer must be negative. The answer is C.

By the way, this turns out to be a geometric sequence because the sequence is created by multiplying each number by the same constant, over and over again. The constant ratio of one term to each term before it is $- 2$ . More on this below.

Definitions

Term: A term is simply one of the numbers in the sequence. Like an algebraic term such as x in the expression $x + 2 y$ , a term is the smallest unit in a sequence and must be understood separately from, but in relation to, the other terms.
Integer: An integer is a number on the real number line such as $- 5$ , $0$ , or $15$ . Fractions such as $\frac{1}{2}$ and $- \frac{1}{3}$ are not integers, nor are decimals such as $- 0.6$ and $1.7$ . Most CLT sequence questions will deal in integers, but some will introduce relationships between the terms that require you to understand how fractions and decimals add to or multiply by each other.
Arithmetic Sequence: In an arithmetic sequence, each term is created by adding a constant amount to the previous term. That number consistently added may be negative, causing the sequence to decrease from left to right. Examples of arithmetic sequences include

$2$ , $4$ , $6$ , $8$ and $0$ , $- 3$ , $- 6$ , $- 9$

Geometric Sequence: In a geometric sequence, each term is created by multiplying the previous term by a constant amount. That number consistently multiplied may be negative, causing the sequence to alternate between and positive and negative values, or a fraction such as $\frac{1}{2}$ , causing the (absolute) value of the sequence to consistently decrease… Examples of geometric sequences include:

$2, 4, 8, 16$ ,
$3, - 6, 12, - 24$
$18, 9, \frac{9}{2}, \frac{9}{4}$ .

Topics for Cross-Reference

Factors and Multiples questions

Variations

Sometimes, instead of providing three or four numbers in a row and asking the student to supply the next number, the CLT will ask for a missing number within a sequence–say, the second number (after supplying the first, third, and fourth numbers).

Also, you may encounter a sequence that combines the features of arithmetic and geometric sequences. Consider the following sequence:

$1, 3, 7, 15, 31...$

There are two ways the think about this sequence. The first is to notice that the difference between the two amounts is a geometric sequence: $2$ , $4$ , $8$ , $16$ , etc. So the next “jump” should be $32$ and therefore the next term is $63$ .

If you are more algebraically inclined, you could note that the sequence follows the expression $2 x - 1$ if we first plug in $1$ for $x$ , then $2$ , etc. This can be difficult to see, but perhaps more easily realized if you realize that each of the numbers is exactly one less than a perfect square.

Strategy Insights

Always determine the pattern. You can do this systematically by running through the three times of sequences: 1) Is it adding the same number each time (keep in mind the number may be negative, in which case it will look like subtraction)? If so, then the sequence is arithmetic. 2) Is it multiplying by the same number each time (knowing that the number multiplied may be negative or a non-integer like a fraction or decimal)? If so, then the sequence is geometric. If neither of these things is true, the sequence is in the “miscellaneous” category, but there is still a pattern to be discovered such as, for instance, a series of perfect squares.
As always, apply the UnCLES method step of looking at the answer choices. Even if you don’t see the pattern right away, you might be able to identify it by working backward from the answer choices.

Flashcard Fodder

The best ways to prepare the foundational skills for sequence questions is to 1) know your multiplication facts and 2) understand how fractions add and multiply with each other.

Sample Questions

Difficulty 1

What is the next term in the arithmetic sequence below?

$10$ , $16$ , $22$ , $28$ , ?, …

A. $30$
B. $34$
C. $36$
D. $38$

(spoiler)

The answer is B. This is an arithmetic sequence where each term equals the previous term plus 6. Or, put differently, if you subtract 6 from a term, you get the term before it. Following this pattern, $34$ must be the next term, because $34 - 6 = 28$ .

Difficulty 2

What is the next term in the geometric sequence below?

$5$ , $25$ , $125$ , ?, …

A. $250$
B. $375$
C. $625$
D. $725$

(spoiler)

The answer is C. This is a geometric sequence where each term equals the previous term times 5. Or, put differently, if you divide a term by 5, you get the term before it. Following this pattern, $625$ must be the next term, because $625 \div 5 = 125$ .

Difficulty 3

What is the next term in the following sequence?

$121$ , $169$ , $225$ , $289$ , ?, …

A. $324$
B. $345$
C. $361$
D. $389$

(spoiler)

The answer is C. The most difficult sequence questions on the CLT present a pattern that is hard to discover. You can rest assured that there is a pattern to discover and, time permitting, set to work on the puzzle. The better your number sense, the more likely you’ll see it. What do, say, 121 and 225 have in common? You could start with “odd”, but can you dig deeper? We encourage CLT to memorize their perfect squares up to $1 5^{2}$ at least, given that a calculator is not permitted. If you’ve done so, you’ll see perfect squares emerge with the first three terms: first $1 1^{2}$ , then $1 3^{2}$ , then $1 5^{2}$ . Sure enough, the next term, $289$ , turns out to be $1 7^{2}$ . Following the pattern, the next number should be $1 9^{2}$ . But which answer is that? As is often the case, we can use reasoning to narrow down the choices. It can’t be an even number, so not A. The remaining three numbers are odd, but think what happens if you multiply $19$ by $19$ by hand. The $9$ ’s get multiplied first, resulting in $81$ . That units digit of $1$ must remain in the final answer. $361$ must be it.

Difficulty 4

Which of the following could be the missing term in the sequence below?

$\frac{1}{2}$ , $x$ , $\frac{1}{5}$ , $\frac{1}{8}$ , $\frac{1}{12}$ , …

A. $1/3$
B. $2/5$
C. $1/4$
D. $1/6$

(spoiler)

The answer is A. Since the missing term is in the middle of the sequence rather than at the end, we have to pay attention to what comes before and after it. The pattern may be hard to discern because the denominators are not changing by a consistent amount; for example, there is a place where the denominator grows by $3$ ( $5$ to $8$ ), then by $4$ ( $8$ to $12$ ). But therein lies the pattern! If it grows by $3$ then by $4$ , we can guess that before that sequence it might grow by $2$ , and before that sequence it might grow by just $1$ . Let’s try it, working backward. If we subtract $2$ from the denominator in the third term, we get $3$ , so the second term would be $\frac{1}{3}$ . That’s answer $A$ . To confirm, we subtract only $1$ from the denominator as we continue to move right to left, and that gives us $\frac{1}{2}$ .

You can also work backward from the answers and use process of elimination. Answer B doesn’t seem to fit because it has a different numerator than all the others. Answer D would actually be smaller than the term after it, which doesn’t make sense since the sequence is decreasing in value. That leaves answer C, which is close, but if we plug it in, the pattern wouldn’t work because the denominator would change by 2, then by 1, then by 3. There is no progression. That leaves answer A$.

Difficulty 5

Which of the following is the missing term in the geometric sequence below?

$\frac{9}{2}$ , $2$ , $x$ , $\frac{32}{81}$ …

A. $8/9$
B. $2/3$
C. $2/9$
D. $1/18$

(spoiler)

The answer is A. This is a difficult sequence to analyze at first glance, but we get a major hint when the question includes the word “geometric”. This means there is a common ratio between any two consecutive terms, so we can take two consecutive terms and divide the second by the first. Doing that with the first two terms gives us $2 \div \frac{9}{2} = \frac{4}{9}$ . So $\frac{4}{9}$ must be the ratio between each pair of terms. That means we simply need to multiply $2$ by $\frac{4}{9}$ to get our answer of $\frac{8}{9}$ . To check our answer, we can multiply by $\frac{4}{9}$ once again, which should give us the last terms of our sequence, and it does: $\frac{8}{9} \times \frac{4}{9} = \frac{32}{81}$ .

For Reflection

How will you approach sequence questions on test day? Write down at least three takeaways from this module.
Rate the difficulty of these questions for you from 1 (no problem) to 5 (problem!). This will help you decide when to answer them and when to skip them on test day.
Did you have difficulty discovering the pattern while practicing any of the questions in this module? Go back and review those questions, teaching yourself what the pattern was in light of the correct answer.