Sequence problems aren’t incredibly common on the test, but if you’re doing well on the quant section, be prepared to see one. Now, you’ve got a choice: do you want to guess quickly and save time for other, easier topics? Do you want to learn some “test savvy” techniques that will help you with some sequence questions but possibly not all of them? Or do you want to learn how to do these every single time, no matter what?

That isn’t a trick question. Every good business person knows that there’s a point of diminishing returns: if you don’t actually need a 51, then you may study for a lower (but still good!) score and re-allocate your valuable time elsewhere.

Try this GMATPrep® problem from the free test. After, we’ll talk about how to do it in the “textbook” way and in the “back of the envelope” way.

* ”For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by . If T is the sum of the first 10 terms in the sequence, then T is

“(A) greater than 2

“(B) between 1 and 2

“(C) between 1/2 and 1

“(D) between 1/4 and 1/2

“(E) less than 1/4”

First, let’s talk about how to do this thing in the “textbook math” way. If you don’t want to do this the textbook math way, feel free to skip to the second method below.

Textbook Method

If you’ve really studied sequences, then you may recognize the sequence as a particular kind called a Geometric Progression. If not, you would start to find the terms and see whether you can spot a pattern.

Plug in k = 1, 2, 3. What’s going on?

What’s going on here? Each time, the term gets multiplied by -1/2 in order to get to the next one. When you keep multiplying by the same number in order to get to the next term, then you have a geometric progression.

This next part gets into some serious math. Unless you really just love math, I wouldn’t bother learning this part for the GMAT, because there’s a very good chance you’ll never need to use it. But, if you want to, go for it!

When you have a geometric progression, you can calculate the sum in the following way:

Next, you’re going to multiply every term in the sum by the common ratio. What’s the common ratio? It’s the constant number that you keep multiplying each term by to get the next one. In this case, you’ve already figured this out: it’s – 1/2.

If you multiply this through all of the terms on both sides of the equation, you’ll get this:

Does anything look familiar? It’s basically the same list of numbers as in the first sum equation, except it’s missing the first number, 1/2. All of the others are identical!

Subtract this second equation from the first:

The right-hand side of the equation is always going to be just the first term of the original sum. The rest of the terms on the right-hand side of the two equations are identical, so when you subtract, they become zero and disappear.

Solve for s:

This value falls between 1/4 and 1/2, so the answer is (D).

Back of the Envelope Method

There is another way to tackle this one. At the same time, this problem is really tricky—so this solution is still not an “easy” solution. Your best choice might be just to guess and move on.

Before you start reading the text, take a First Glance at the whole thing. It’s a problem-solving problem. The answers are… weird. They’re not exact. What does that mean?

Read the problem, but keep that answer weirdness in mind. The first sentence has a crazy sequence. The question asks you to sum up the first 10 terms of this sequence. And the answers aren’t exact… so apparently you don’t need to find the exact sum.

Take a closer look at the form of the answers. Notice anything about them?

They don’t overlap! They cover adjacent ranges. If you can figure out that, for example, the sum is about 3/4, then you know the answer must be (C). In other words, you can actually estimate here—you don’t have to do an exact calculation.

That completely changes the way you can approach this problem! Here’s the sequence:

According to the problem, the 10 terms are from k = 1 to k = 10. Calculating all 10 of those and then adding them up is way too much work (another clue that there’s got to be a better way to do this one). So what is that better way?

Since you know you can estimate, try to find a pattern. Calculate the first two terms (we had to do this in the first solution, too).

What’s going on? The first answer is positive and the second one is negative. Why? Ah, because the first part of the calculation is -1 raised to a power. That will just keep switching back and forth between 1 and -1, depending on whether the power is odd or even. It won’t change the size of the final answer, but it will change the sign.

Okay, and what about that second part? it went from 1/2 to 1/4. What will happen next time? Try just that part of the calculation. If k = 3, then just that part will become .

Interesting! So the denominator will keep increasing by a factor of 2: 2, 4, 8, 16 and so on.

Great, now you can write out the 10 numbers!

… ugh. The denominator’s getting pretty big. That means the fraction itself is getting pretty small. Do I need to keep writing these out?

What was the problem asking again?

Right, find the sum of these 10 numbers. Let’s see. The first number in the sequence is 1/2 and the second is -1/4, so the pair adds up to 1/4.

Right now, the answer would be right between D and E. Does the sum go up or down from here?

The third number will add 1/8, so it goes up:

But the fourth will subtract 1/16 (don’t forget that every other term is negative!), pulling it back down again:

Hmm. In the third step, it went up but not enough to get all the way to 1/2. Then, it went down again, but by an even smaller amount, so it didn’t get all the way back down to 1/4.

The fifth step would go up by an even smaller amount (1/32), and then it would go back down again by yet a smaller number (1/64). What can you conclude?

First, the sum is always growing a little bit, because each positive number is a bit bigger than the following negative number. The sum is never going to drop below 1/4, so cross off answer (E).

You keep adding smaller and smaller amounts, though, so if the first jump of 1/8 wasn’t enough to get you up to 1/2, then none of the later, smaller jumps will get you there either, especially because you also keep subtracting small amounts. You’re never going to cross over to 1/2, so the sum has to be between 1/4 and 1/2.

The correct answer is (D).

As I mentioned above, you may decide that you don’t want to do this problem at all. These aren’t that common—many people won’t see one like this on the test. Also, you don’t have to get everything right to get a top score. Just last week, I spoke with a student who outright guessed on 4 quant problems, and she still scored a 51 (the top score).

Key Takeaways for Advanced Sequence Problems

(1) Do you even want to learn how to do these? Don’t listen to your pride. Listen to your practical side. This might not be the best use of your time.

(2) All of these math problems do have a textbook solution method—but you’d have to learn a lot of math that you might never use if you try to learn all of the textbook methods. That’s not a problem if you’re great at math and have a great memory for this stuff. If not…

(3) … then think about alternate methods that can work just as well. Certain clues will indicate when you can estimate on a problem, rather than solving for the “real” number. You may already be familiar with some of these, for instance when you see the word “approximately” in the problem or answer choices that are spread pretty far apart. Now, you’ve got a new clue to add to your list: answers that offer a range of numbers and the different answer ranges don’t overlap.

* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.

So your Critical Reasoning (CR) score has moved a little, but not enough. Or each question is still taking you 3 minutes to answer. You’ve studied for months, read the Strategy Guides, taken every practice test, and completed every Critical Reasoning question in the big Official Guide and the Verbal Review supplement so many times you have them all memorized. What more can you do? Do more questions? You can probably imagine, more questions will usually mean more of the same issues, and simply reinforce bad habits…

Chances are, despite all your hard work, you’re still using your intuition and “gut feeling” to answer CR questions. Unfortunately, your gut feeling works some of the time, but not 100% of the time. Remember, the test is designed so that the average person picking what “looks right” will get only 50% of the questions correct.

So what to do? For now, stop doing more questions until you 1) learn the formal rules of logic behind how CR works, and 2) deeply analyze all the questions you’ve done for repeating patterns: question types, patterns of reasoning, logical flaws, right and wrong answer types, etc.

So that’s what the next few weeks will be about. Each week, I’ll post an article that goes absurdly in-depth about one aspect of the logic behind CR, along with exercises to apply those lessons. These are the same exercises I do with my tutoring students, who have found them very effective. I’m also interested in your feedback: what worked for you? What didn’t? Questions and concepts you’re still struggling with? I’m open to discussion and debate.

So let’s get started. I’ll start with the essentials and then really nerd out on formal logic, so keep reading to the end.

LESSON ONE: RTFQ

In our classes, we teach a four-step process to answering CR questions:

1) Identify the question (Know what the question is asking and what kind of question it is)
2) Deconstruct the argument (Analyze each piece of the passage for what role it plays)
3) Pause and state the goal (Predict what the correct answer should do)
4) Work from wrong to right (Use process of elimination to get to the right–or “least wrong”–answer.)

Today’s focus: Step one, which I call RTFQ, as in “Read the F___ Question” (F as in Full! Read the Full question. What were you thinking?)

The basics: The GMAT only asks a limited number of questions, with very rare variation. Each type of question implies HOW you should deconstruct the argument and WHAT the right answer will do. If you don’t identify the question properly, you won’t look for the right things, or you’ll waste time reading for things that aren’t there. So…Right now, can you name them all? No really.

Exercise 1: Before you scroll down, get out your notebook and write down as many types of questions as you can think of. Ready? Go.
.
.
.
.
.

How many did you come up with? 5? 6? Depending on how you break them down (what books you’ve read and who taught you), there are anywhere from 10-13 common types of questions. Here’s a list of the 11 most common that I use, grouped by category.

Structure based:
Identify the bolded part (role in the reasoning)
Identify the overall reasoning
Identify the conclusion
Mimic the reasoning (also known as parallel the reasoning)

Reasoning/assumption based:
Assumption
Strengthen
Weaken (and Flaw questions)
Evaluate
Fill in the Blank

Evidence or fact-based questions:
Inference (also known as “Draw a Conclusion” questions)
Resolve or Explain (a paradox or discrepancy)

I’ll explain more about the categories in future articles, but for now… Can you identify them when they show up? One of the most common mistakes you can make on the GMAT is simply misidentifying the question (e.g. mistaking strengthen for inference or strengthen for explain).

Exercise 2: Pick a dozen questions and name them! Take out your Official Guide for GMAT Review and get to work. Let’s pick numbers 50 through 61. If you need help, skim the passage itself to (All questions excerpted from The Official Guide for GMAT Review 13th Edition, by GMAC®)

50. Which of the following, if true, most strengthens the argument above?
51. The argument is most vulnerable to the objection that it fails to
52. Which of the following, if true, most strongly supports Summit’s explanation of its success in retaining employees?
53. Which of the following strategies would be most likely to minimize company X’s losses on the policies?
54. If the statements above are true, which of the following must be true?
55. Which of the following most logically completes the argument given below?
56. The conclusion above would be more reasonably drawn if which of the following were inserted into the argument as an additional premise?
57. Which of the following, if true, most helps to explain the surprising finding?
58. Which of the following, if true, most seriously weakens the conclusion above?
59. Which of the following most logically completes the passage?
60. If the facts stated in the passage above are true, a proper test of a country’s ability to be competitive is its ability to
61. Which of the following, if true, does most to explain the contrast described above?

And for good measure, identify number 66.

66. Which of the following conclusions about Country Z’s adversely affected export-dependent industries is best supported by the passage?

Write down what you think each one is.

Ready for your answers? Here’s how I identified them, with commentary.

50. Strengthen: Pretty straight up. The correct answer will strengthen the argument above.
51. Weaken: Or more specifically, identify the flaw in the reasoning. The words “it fails to” mean that the right answer, when considered, will damage the argument.
52. Strengthen: Don’t let the word “explain” fool you. The explanation is already in the argument; in fact the explanation may be the conclusion of the argument. Your job is to find an additional piece of evidence to strengthen that explanation.
53. Resolve/Explain: This one was tough. The question implies that there’s a problem (losses) to be solved (“minimize[d]”), which is what many resolve/explain questions do. Also, the argument itself describes a pretty clear contradiction: how does X keep its prices low, but also make enough income to pay for claims? The answer will resolve this. Feel free to argue with me in the comment section, though.
54. Inference (also known as “draw a conclusion”): Notice how “the statements above are true.” That mean you WON’T be looking for premises and conclusions, just putting facts together to find out what else must be true. More about this later in the section about “Deductive Reasoning.”
55. Fill in the Blank: note that the blank part starts with the word “because____” so you’ll be providing a premise that helps the conclusion. So, in a way, you can look at this as a strengthen question, too.
56. Assumption: Yes, assumption, though if you named this as a strengthen question, you’ll probably get it right. Technically, though, when the GMAT asks for an additional or unstated premise that makes the argument “more reasonably drawn” or that is “required,” it’s asking you for the assumption. But it’s interesting to note that assumption and strengthen questions both do the same thing: support the reasoning of an argument.
57. Resolve/explain: NOT strengthen. Imagine walking into your house to find your favorite chair is broken. Explaining WHY it’s broken is far different from Strengthening or fixing the chair with additional support.
58. Weaken: fair enough, easy to spot.
59. Fill in the blank: and with the word “since____” leading off the blank, it’s another strengthen.
60. Inference: Again, “if the statements above are true…” your reading for facts, not arguments.
61. Resolve/explain: again

aaaaand #66?

66. Inference: Yes. Inference. NOT STRENGTHEN! For more about how to differentiate between Inference and Strengthen questions, see our Critical Reasoning Strategy Guide, chapter 6.

So, how’d you do? If you were less than 100%, spend some time with the strategy guide, focusing on how to identify question types. Write down several examples of each question type and quiz yourself some more. You can use the Official Guide Problem Sets in the back of the CR Strategy Guide to see whether you were right or not. Keep working until you’re 100%.

NERDING OUT ON LOGIC

Critical reason is a test of LOGIC. So, with a big stack of logic books next to me, I’m going to discuss some of the formal rules behind the what GMAT writes questions. Ready?

The GMAT uses the word conclusion in two different ways. Most of the time, the GMAT is referring to an “inductive” conclusion, but occasionally, it’s asking about a “deductive” conclusion. Don’t confuse them!
So to explain: There are two kind of reasoning in the word: deductive reasoning and inductive reasoning.

Deductive reasoning is more concrete, more mathematical, more “true.”

Wikipedia’s definition of deductive reasoning: “Deductive reasoning links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true.”

In other words, if the premises are true, then the conclusion must be true. Does this sound like a common question type? (Hint: it starts with an “I____”)

Here are some examples of deductively valid arguments.

Premise: Sally is taller than Frank.
Premise: Frank is taller than William
Conclusion: Sally must be taller than William.
(Other deductively valid conclusions: Frank is shorter than Sally. William is not the same height as sally.)

Premise: All cats are persnickety
Premise: Mr. Whiskers is a cat.
Conclusion: Mr. Whiskers is persnickety.
(Other deductively valid conclusions: Some persnickety things are cats. At least one cat is named Mr. Whiskers.)

Inductive reasoning is a little softer, and much more common on the GMAT and in the real world. Science, economics, medicine, and our justice system are largely based on induction.

Wikipedia’s definition again: “Inductive reasoning is reasoning in which the premises seek to supply strong evidence for (not absolute proof of) the truth of the conclusion. While the conclusion of a deductive argument is supposed to be certain, the truth of an inductive argument is supposed to be probable, based upon the evidence given.”

In other words, if the premised are true, then the conclusion has a probability of being true, but also a probability of being false.

I’m usually sleepy after 11:00pm.
It’s past midnight.
I must be sleepy.

3 out of 4 dentists recommend chewing OctiDent after meals.
You should chew Octident after every meal.

After I cut bacon out of my diet, I lost 5 pounds.
If you want to lose weight, you should cut bacon out of your diet.

Inductively valid arguments have a very high probability of being true, with little chance of contradictory evidence (good scientific theories). Inductively invalid arguments have a high probability of being false (horoscopes). The dividing line between valid and invalid arguments can be shady and can depend on context. 90% certainty would be a great bet at a casino, but a lousy bet on airplane guidance systems.

We’ll get more into how to evaluate inductive reasoning vs. deductive reasoning in later articles, but for now, lets just learn to spot it.

Exercise: peruse the Official Guide questions 50-61 again. Decide whether the question and argument will be based on induction or deduction (Hint: if the argument can be helped or hurt, it’s probably induction. In the conclusion must be true, it’s deduction.)

50. Which of the following, if true, most strengthens the argument above?
51. The argument is most vulnerable to the objection that it fails to
52. Which of the following, if true, most strongly supports Summit’s explanation of its success in retaining employees?
53. Which of the following strategies would be most likely to minimize company X’s losses on the policies?
54. If the statements above are true, which of the following must be true?
55. Which of the following most logically completes the argument given below?
56. The conclusion above would be more reasonably drawn if which of the following were inserted into the argument as an additional premise?
57. Which of the following, if true, most helps to explain the surprising finding?
58. Which of the following, if true, most seriously weakens the conclusion above?
59. Which of the following most logically completes the passage?
60. If the facts stated in the passage above are true, a proper test of a country’s ability to be competitive is its ability to
61. Which of the following, if true, does most to explain the contrast described above?

Answer Key:

50. Induction
51. Induction
52. Induction
53. Induction
54. DEDUCTION
55. Induction
56. Induction
57. Induction (The explanation will be inductively valid.)
58. Induction
59. Induction
60. DEDUCTION
61. Induction

What do you think about question 66? Discuss and debate it in the comments below!

Take some time looking up deductive reasoning vs. inductive reasoning on the web. Wikipedia is a good place to start. Then, start analyzing other questions for the kind of reasoning tested on each. You may find that a lot of the questions you got wrong were one type or the other.

For an advanced drill, dig up all the Inference questions you can find. (I’ll give you a few: 66, 91, 103, and 104) Some of them are asking you for deductively valid conclusions, while others are asking for inductively valid conclusions. Can you determine which is which? Again, post your results in the comments section below.

Get to work, and for now just focus on those questions! See you in future articles.

Below is an excerpt from Andrew Yang‘s new book, Smart People Should Build Things: How to Restore Our Culture of Achievement, Build a Path for Entrepreneurs, and Create New Jobs in America, which comes out in February 2014. Andrew was named Managing Director of Manhattan GMAT in 2006, Chief Executive Officer in 2007, and President in 2010. He left Manhattan GMAT in 2010 to start Venture for America, where he now serves as Founder and CEO. 

Professional Services as a Training Ground.  

As we’ve seen, one of the most frequently pursued paths for achievement-minded college seniors is to spend several years advancing professionally and getting trained and paid by an investment bank, consulting firm, or law firm. Then, the thought process goes, they can set out to do something else with some exposure and experience under their belts.  People are generally not making lifelong commitments to the field in their own minds. They’re “getting some skills” and making some connections before figuring out what they really want to do.

I subscribed to a version of this mind-set when I graduated from Brown. In my case, I went to law school thinking I’d practice for a few years (and pay down my law school debt) before lining up another opportunity.

It’s clear why this is such an attractive approach. There are some immensely constructive things about spending several years in professional services after graduating from college. Professional service firms are designed to train large groups of recruits annually, and they do so very successfully. After even just a year or two in a high-level bank or consulting firm, you emerge with a set of skills that can be applied in other contexts (financial modeling in Excel if you’re a financial analyst, PowerPoint and data organization and presentation if you’re a consultant, and editing and issue spotting if you’re a lawyer). This is very appealing to most any recent graduate who may not yet feel equipped with practical skills coming right out of college.
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Did you know that you can attend the first session of any of our online or in-person GMAT courses absolutely free? We’re not kidding! Check out our upcoming courses here.

How do you study? More importantly, how do you know that the way in which you’re studying is effective—that is, that you’re learning what you need to learn to improve your GMAT score? Read on! Read more

Did you know that you can attend the first session of any of our online or in-person GMAT courses absolutely free? We’re not kidding! Check out our upcoming courses here.

Whether you’ve been studying for a while or are just getting started, let’s use the New Year as an opportunity to establish or renew your commitment to getting your desired GMAT score. Read more

We invite you to test your GMAT knowledge for a chance to win! The second week of every month, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that month’s drawing for free Manhattan GMAT prep materials. Tell your friends to get out their scrap paper and start solving!

Here is this month’s problem:

If p, q, and r are different positive integers such that p + q + r = 6, what is the value of x ?

(1) The average of x^p and x^q is x^r.

(2) The average of x^p and x^r is not x^q.

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Below is an excerpt from Andrew Yang‘s new book, Smart People Should Build Things: How to Restore Our Culture of Achievement, Build a Path for Entrepreneurs, and Create New Jobs in America, which comes out in February 2014. Andrew was named Managing Director of Manhattan GMAT in 2006, Chief Executive Officer in 2007, and President in 2010. He left Manhattan GMAT in 2010 to start Venture for America, where he now serves as Founder and CEO. 

The Prestige Pathways Part II.  

You could ask, so what if our talented young people all march off to become lawyers, doctors, bankers, and consultants? Isn’t that what smart people are supposed to do?

There are a few problems with this stance. First, the degree to which the recruitment infrastructure exists is a relatively recent phenomenon. Bain and Company, a premier management consulting firm, wasn’t founded until 1973—now it employs over 5,000 talented people and recruits hundreds per year. The financial services industry has mushroomed in size, with Wall Street firms employing 191,800 at their peak in 2008, up from only 65,300 in 1975. The growth in professional services has given rise to an accompanying set of recruitment pipelines only in the past several decades.

Yet the allocation of talent is a zero-sum game. If the academically gifted are funneled in higher numbers toward finance and consulting, then lesser numbers are going into other areas, such as the operation of companies, startups, and early-stage enterprises. In the United States, companies with fewer than 500 employees account for almost two-thirds of net new jobs and generate thirteen times more new patents per employee than do large firms. If the US economy had generated as many startups each year for 2009–12 as it had in 2007, the country would have produced almost 2.5 million new jobs by 2013. If we’re interested in spurring long-term job growth, we want as much talent as possible heading to new firms so that more of them can succeed, expand, and hire more people.
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