Textbook

*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.

What is the next term in the sequence below?

-4, 8, -16, 32, ?, …

A. 64

B. -48

C. -64

D. -96

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.

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 $2x−1$ 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.

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.

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 $34−6=28$.

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 $625÷5=125$.

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 $15_{2}$ 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 $11_{2}$, then $13_{2}$, then $15_{2}$. Sure enough, the next term, $289$, turns out to be $17_{2}$. Following the pattern, the next number should be $19_{2}$. 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.

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

$21 $, $x$, $51 $, $81 $, $121 $, …

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 $31 $. 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 $21 $.

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**.

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

$29 $,$2$,$x$,$8132 $ …

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÷29 =94 $. So $94 $ must be the ratio between each pair of terms. That means we simply need to multiply $2$ by $94 $ to get our answer of $98 $. To check our answer, we can multiply by $94 $ once again, which should give us the last terms of our sequence, and it does: $98 ×94 =8132 $.

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