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14, 23, 32, 41, 50, 59, …

1) In the sequence above, each term is 9 more than the previous term. What is the 41^{st} term of the sequence?

- 360
- 365
- 369
- 374
- 383

2) What is the difference between the fourth and third terms of the sequence defined by

- 18
- 23
- 47
- 65
- 83

3) Which of the following could be true of at least some of the terms of the sequence defined by

I. divisible by 15

II. divisible by 18

III. divisible by 27

- I only
- II only
- I and II only
- I and III only
- I, II, III

4) Let S be the set of all positive integers that, when divided by 8, have a remainder of 5. What is the 76^{th} number in this set?

- 605
- 608
- 613
- 616
- 621

5) Let T be a sequence of the form . If and , find .

- 37
- 38
- 39
- 40
- 41

6) What is the sum of all the multiples of 20 from 160 to 840?

- 14,000
- 17,500
- 18,060
- 28,000
- 35,000

7) A sequence is defined by for n > 2, and it has the starting values of and . Find the value of .

- 25
- 32
- 36
- 93
- 279

9) In the set of positive integers from 1 to 500, what is the sum of all the odd multiples of 5?

- 10,000
- 12,500
- 17,500
- 22,500
- 25,000

10) If =, , , , and , what is the value of ?

- 7
- 11
- 19
- 42
- 130

Solutions will follow this article.

Sequences are a tricky topic on the GMAT Quant section. In that linked article, I discuss sequence notation, which is a variant of function notation, and I discuss recursive sequences, that is, sequences in which each term is determined by the previous term or terms.

An **arithmetic sequence** is a sequence in which we add some fixed amount to each term to get the next term. Another way to say that is that, if we subtract any term from the following term, the difference will always be the same: this difference is called the **common difference** of the arithmetic sequence. Let d be the common difference. Then, in algebraic form, the terms of the arithmetic sequence would be

Recall that, say, for the 3^{rd} term, the little subscript 3 is the **index**, that is, the position on the list. For this particular sequence, every term equals the first term plus a factor times **d**, and that factor is always one less than the index. This means we can write the general term as:

That is a very important formula, although, as always, don’t just memorize it; instead, remember the logic of the argument that leads up to it.

This is an important formula because any evenly spaced list is an arithmetic sequence. The consecutive multiples of any factor form an arithmetic sequence.

The sum of the first n terms in an arithmetic sequence is given by the formula

That’s the sum of the first term and the last term, times half the number of items on the list. You can also thing of that as the average of the first & last terms times the number of items on the list. One special case is the sum of the first n integer, given by

If you had some insights while reading that first article on sequences or the section on arithmetic sequences, then take another look at the problems above before looking at the solutions below. If you found this article helpful, or if you have an alternative solution for solving any of these problems, please let us know in the comments section below!

1) This is an arithmetic sequence, with

= 14 and d = 9

Using the formula for the nth term, we find that:

= 14 + 9*40 = 14 + 360 = 374.

Answer = **(D)**.

2) This is general sequence, with an explicitly defined nth term.

Answer = **(C)**.

3) First of all, if n = 8, then

= (16 – 1)(16 + 3) = 15*19

We don’t have to calculate that: clearly, whatever it is, it is divisible by 15. Similarly, if n = 12, then

= (24 – 1)(24 + 3) = 23*27

Whatever that equals, it must be divisible by 27. Thus, I & III are true. Notice that, for any integer, 2n must be even, so both (2n – 1) and (2n + 3) are odd numbers, and their product must be odd. Every term in this sequence is an odd number. Now, no odd number can be divisible by an even number, because there is no factor of 2 in the odd number. Therefore, no terms could possibly be divisible by 18. Statement II is absolutely not true.

Answer = **(D)**.

4) Think about the first few numbers on this list:

5, 13, 21, 29, 37, 45, …

Notice that 5 is the first number in S, because when 5 is divided by 8, the quotient is zero and the remainder is 5.

This is an arithmetic sequence with

= 5 and d = 8

Using the general formula for the nth term of an arithmetic sequence, we have

= 5 + 8*(75) = 5 + 4*(150) = 5 + 600 = 605

Answer = **(A)**.

5) The formula in the first sentence tells us that this is an arithmetic sequence. The first term and the common difference are unknown, but we can generate two equations from the values of the two terms given.

Subtract the first equation from the second, and we get 16d = 48, which means d = 3. From the value of third term, we can see that first term must equal 11. Therefore,

Answer = **(B)**.

6) This is arithmetic sequence, and we have the first and last terms already. How many terms are there? Well, 160 = 20*(8) and 840 = 20*(42); we have to use inclusive counting to see that there are 42 – 8 + 1 = 35 terms.

Answer = **(B)**.

7) This is a recursive sequence, and we have to find it term by term.

Answer = **(E)**.

8) This is a recursive sequence, so we have to work backwards term by term.

Answer = **(A)**.

9) Let’s think about the terms in this sequence:

5, 15, 25, 35, …., 485, 495

The first term is 5 and the last is 495. There are 100 multiples of 5 from 1 to 500, so there are 50 odd multiples and 50 even multiple. The sum is:

Answer = **(B)**.

10) This recursive sequence is probably more difficult than anything that the GMAT is going to throw you, but solving this problem is not too bad. As with any recursive sequence, we have to go term by term.

Answer = **(C)**.

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Hello!

Today, it’s fraction time! I am answering a question that one of your fellow GMAT-studiers asked about comparing fractions. Watch the video to learn more on this topic, and check out Mike McGarry’s blog post for even more tips!

If you have any questions about this, be sure to leave them in the comments below.

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*Analytical Writing Assignment:*One essay about the weaknesses in an argument that you read*Integrated Reasoning:*Math-based questions on reading tables, charts, and diagrams*Quantitative:*Math*Verbal:*Reading comprehension multiple choice, grammar, critical reasoning

*Reading:*Reading comprehension multiple choice*Listening:*Listening comprehension multiple choice*Speaking:*Open responses (speaking into a microphone), including summarizing recordings and text*Writing:*One essay summarizing a lecture, another essay explaining your opinion

So although two of the sections on each test are measure the same basic skills (verbal and writing), there are very few questions that are similar. In fact, the only part of either test that’s actually comparable is the reading comprehension.

Both the GMAT and the TOEFL focus on advanced, academic English. Neither test includes fiction, for example, and both are likely to include some text about science or history. However, that is the only real similarity between the two. In fact, the passages are written in a very different style between the two tests, and the questions are not very similar at all.

That’s because they have different goals. The GMAT wants to test your logical abilities and how well you can make concrete inferences from a text without assuming too much. In other words, your goal during GMAT reading comprehension is to understand the author’s purpose and show that you can use information from that text correctly to create your own conclusions.

The TOEFL, on the other hand, only wants you to understand the language. Very much of TOEFL reading comprehension relies on vocabulary. Even questions that are not directly about vocabulary require you to understand difficult sentence structures and idioms, but do not require you to analyze the information you read. You only need to understand it and recognize other sentences that give the same information.

Besides that, TOEFL reading passages, at around 700 words, are twice as long as “long” GMAT passages, which rarely pass 350 words, and there are many more questions per passage (12-14 on the TOEFL, 3-4 on the GMAT) so the strategies for answering questions are different.

These two parts of the tests have some things in common, but only very little.

Basically, the GMAT tests grammar and written style extensively. The TOEFL doesn’t test grammar directly, but if you write and speak with correct grammar, then you’ll score better on those sections. However, the GMAT’s grammar is mostly **very** advanced and often extremely subtle. If you understand English grammar well enough to score well on GMAT sentence correction questions, you’ll probably be able to write well.

And finally, the GMAT and the TOEFL both require you to structure an essay well. Using transition phrases (such as “on the other hand,”) and advanced vocabulary helps on both tests. The GMAT essay is about much, much more than that—your logical analysis of an argument, specifically—but the fundamentals of good English writing apply to both tests.

So even if the question types are different between the two tests, improving your written grammar can help for both the GMAT and the TOEFL.

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*Hear from a Yale grad about his experience with the GMAT. Thanks for sharing, Brendan!*

* About Me:* My name is Brendan and I’m an Electrical Engineering Major from Yale University (2010). I have been a submarine officer in the US Navy for the past four years. My job is primarily concerned with operating the reactor on board, so I am looking to pursue energy marketing and infrastructure at business school.

* Biggest Challenge:* I won’t lie, despite my engineering background, I was still weak on many quantitative skills required by the GMAT, and was also very weak on sentence correction (not surprising considering years of short hand engineering prose). I attacked this initially by skimming the lessons and really attacking practice problems with volume. In the end I realized two things:

1. Many of the lessons that seem simple have very valuable parts crucial to getting top GMAT scores. It wasn’t enough to skip the areas I thought I already understood. **Take the time to expose yourself to all the information in the lessons.**

2. Practice problems are only effective with a **thorough review**. Taking the time to understand why I got something incorrect (or often why I got something correct) was far more valuable than continuing to be incorrect on similar questions. Magoosh’s explanation videos are terrific and proved to be a great asset once I decided to go through them more thoroughly.

As my quantitative improved, I shifted my focus to my weaker verbal section. I poured in time with lessons and practice questions. In the end, I did very well in verbal (99 percentile), but actually a little worse than I expected in math (74 percentile). My first practice test with Magoosh was a 630, so I was happy with my overall score of 760 – but it was clear that I neglected my math studies during my last big push in verbal.

* Tips for Others:* Remember: if an engineer like me can beat the verbal section, so can you, but

Hello!

Welcome to another GMAT Tuesday installment, complete with a 2-second guitar solo! This week, learn about two tricky exceptions to the “empty it” rule that might trip you up on test day.

If you have any questions about this, please leave them in the comments below.

]]>1) At a certain school of 200 students, the students can study French, Spanish, both or neither. Just as many study both as study neither. One quarter of those who study Spanish also study French. The total number who study French is 10 fewer than those who study Spanish only. How many students study French only?

- 30
- 50
- 70
- 90
- 120

2) In a company of 300 employees, 120 are females. A total of 200 employees have advanced degrees, and the rest have a college degree only. If 80 employees are males with college degree only, how many are females with advanced degrees?

- 60
- 80
- 100
- 120
- 160

3) In a certain school, there are 80 Freshmen, 100 Sophomores, and 220 Upperclassmen, drawn from three cities: A, B, and C. Sixty percent of students are from A, 30% from B, the rest from C, and all these students from C are freshmen. Half the student from B are upperclassman, and the rest are split evenly between the other two grades. How many sophomores are from A?

- 60
- 70
- 80
- 90
- 100

4) There are a total of 400 students at a school, which offers a chorus, baseball, and Italian. This year, 120 students are in the chorus, 40 students in both chorus & Italian, 45 students in both chorus & baseball, and 15 students do all three activities. If 220 students are in either Italian or baseball, then how many student are in none of the three activities?

- 40
- 60
- 70
- 100
- 130

Solutions will be given at the end of the article.

For two or more overlapping categories, in scenarios in which any given member can belong to all, some, or none of the categories, Venn Diagrams can be helpful. If there are two separate variables, and each member is classified according to each variable, then the Double Matrix Method can be helpful. If reading the articles at either of those links gives you insights on any of these problems, you may want to give the problems a second look before reading the solutions below.

1) Let x be the number of folks studying both, which means it is also the number of folks studying neither.

“One quarter of those who study Spanish also study French.” If the Spanish students studying French are x, then all Spanish students are 4x, and those who do not study French are 3x. Also, let y be the number of students who study French but not Spanish.

“The total number who study French is 10 fewer than those who study Spanish only.” In other words,

x + y = 3x – 10

10 = 2x – y

Also, notice that the total number of students is 200:

3x + x + y + x = 200

5x + y = 200

We have two equations with two unknowns. Add the equations (2x – y = 10) and (5x + y = 200), and we get

7x = 210

x = 30

y = 50

And the number who study French is x + y = 80. Answer = **(B)**

2) This problem is handled best with a double-matrix solution.

Because there are 120 females, we know the rest are males. Because there are 200 with advanced degrees, we know there are 100 with college degrees only.

At this point, we could go either way: we could figure out all the numbers in the “college” row or all the numbers in the “male” column.

Either way, it is easy to fill in the last box:

Thus, there are 100 employees who are females with advanced degrees. Answer = **(C)**.

3) This problem calls for a 3×3 double matrix. Here are the numbers for the grade levels, with the column totaled:

Obviously, 10% of 400 is 40, so 30% is 3*40 = 120, and 60% = 6*40 = 240. This allows us to complete the bottom row.

All 40 from C are freshmen, so we can put zeros in the other two grade level slots:

Half of the 120 from B, 60, are upperclassmen, and the other 60 are evenly split, 30 in each of the other two slots.

Now, we can complete the sophomore row:

There are 70 sophomores from A. Answer = **(B)**.

4) This problem calls for a 3-way Venn Diagram. Here’s the diagram with no numbers filled in.

We know that C = 15. If 40 students are in both chorus & Italian, B + C = 40, and because C = 15, B = 25. If 45 students in both chorus & baseball, C + F = 45, and F = 30. We know that there are 120 in chorus, and B + C + F = 70, so E = 50.

Now, we are told that 220 student are in either Italian or baseball. Think about that region, Italian or baseball:

That entire purple region, A + B + C + D + F + G, is 220. If we add E = 50, that’s a total of 270 inside all three circles, which means that the outside of the circle, H, must equal 400 – 270 = 130. Answer = **(E)**.

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One place in particular where our focus can wain is in the Verbal section. There is so much to read, and so much of it seems uninteresting. But the moment you let that idea slip into your mind, you’ve lost the battle. You need to have a strategy for staying focused and interested, even during the most banal passages.

Let’s look at some ways to stay focused and engaged.

Don’t let the words just wash over you when you read. Engage with them and ask questions about what you are reading. The key is to continually ask questions, to ask the same questions, and change your answer as you read more. Iterate through a set of questions and update the answer as you encounter more information.

For example, you should ask yourself: What’s the purpose of this passage? What’s the main idea? Where’s this going? How does the author feel about all this? Use these questions as a basis for investigating this unknown topic. At first you may want to have these written on a slip of paper to refer to every time you read something—not just passages on the GMAT. As you practice, these questions will be imprinted in your mind, and your reading, out of habit, will involve answering these questions.

Pens aren’t just for writing! Use the pen to keep track of what you are reading. Follow each line with the pen. It gives your hand something to do and will keep you focused on pushing forward and moving through the passage.

This may seem outdated and something you did when you were just learning to read. And it probably was, but that doesn’t mean it won’t help you to stay focused. By using a pen and moving it across the screen or page, you make reading a physical act. As such, you will engage more with what you are reading.

If you’re holding a pen, you might as well put it to good use. But don’t just write down things when you see something in the passage that sparks your attention. Rather, focus on taking notes on the flow of the passage, the major points and shifts in tone in the passage.

For example, the first paragraph will introduce some idea, generally, and you can make note of what that is. Then in the next paragraph, there are a few things that might happen. The author might reverse directions and argue against that idea, or she might present a counterpoint of view. Or, the next paragraph could easily be an example of what was just introduced, give supporting evidence, or move on and talk about the result of what was just mentioned.

You should keep track of this flow and change. Don’t worry too much about writing down what is in the paragraph. Rather focus on the purpose of the paragraph or the purpose of a section.

Of all the things you could do, the most important is to be curious. You need to cultivate a general curiosity, which is not easy to do, but will help you so much on the test. If you approach something and your first reaction is, “This is ridiculous! Who wrote this?!? Who cares about this stuff?” You will struggle through the entire test, and you will quickly lose focus.

But if you approach a passage, and your reaction is, “Woah! That’s weird! Why’s that?” you will find the test much easier and less like work. Remember that the passages you read, although esoteric, come from actual articles and journals. The information does have a connection to our world. And if you can dive into the test, open to learning something new, if you are up for a challenge and down to solve some puzzles, you will dominate the test and never lose focus.

No one comes into this world born to focus. Focus is something cultivate. We have to train ourselves to focus. If you want to stay focused, and you want to improve your score, you will need to take these tips and start applying them. Don’t expect them to work right away. Adapt these tips as you develop and grow. Over time you will see the benefit.

]]>Hello!

Meet my friend Ocky and get up to speed on participles! This week’s video is an addition to the previous video on participles, so I hope this helps answer more questions you might have about the topic. And if you’d like to learn even more, be sure to read this post for more details.

Here’s a still of the final board work:

If you have any questions, comments or suggestions for future videos, please share them in the comments below!

]]>While students’ ambitions are commendable, and while curiosity about this point is eminently understandable, there is something inherently flawed in the logic of that question. This requires a slightly deeper examination.

What does “700+ level” mean on the GMAT? Well, if we look at GMAT Percentiles, the percentile for 700 is 89%, which roughly means that 10-11% of the population scores at that level or above, and 89% of the population scores below that level. If we were to sample a large number of GMAT test takers, ask them all a particular question, and fewer than 10% could get the question correct, then with some justification some might refer to this as a 700+ level question. Notice that a very large process of data collection would be needed to verify that any question is of this sort.

Even if we were to gather all that data, the notion is still suspect. Suppose we were to gather a large batch of questions that 90% of the test-taking population regularly get wrong. The fact that one can get a single question of this sort correct doesn’t mean bupkis. If you can get these very difficult questions correct 30-50% of the time, that’s not necessarily telling. If you can simply nail them almost every single time, that’s impressive, and certainly would be more indicative of an elite level of performance.

Even then, it’s a mistake to think of the test too reductionistically, as if mastery of each individual possible question necessarily leads to mastery of the test. This overlooks the holistic quality of excellence that needs to pervade your work to produce an excellent performance on the GMAT. Similarly, it overlooks the all-too-common problem of a precipitous drop from performance on practice exams to the performance on test day. The perspective and mindset and focus and lack of stress that you bring to the test are far more important than the skills to perform on any particular question.

Finally, of course, remember that the Quant and Verbal sections are subject to Computer Adaptive Testing. The profound implications of this are little understood and little appreciated. As you sit for the real GMAT, and get question correct or incorrect, the test will use a complex psychometric procedure to increase or decrease the difficulty of the subsequent questions you receive, and your final grade is determined by some similarly sophisticated and abstruse algorithm. The paradox is: a person who gets a 550 and a person who gets a 750 may get approximately the same *number* of questions correct and incorrect, but the *difficulty* of the correct and incorrect questions will differ significantly. The nature of this algorithm is strictly the propriety knowledge of GMAC, so even if I were to have an expertise in psychometrics, I would not access to the details of how that algorithm works. Thus, in a profound sense, we have essentially no way of exactly knowing how the relative difficulty of any particular question influences one’s GMAT score. In this deepest sense, given all publically available knowledge, the idea of a “700+ question” is entirely meaningless and without basis in objective reality.

I say that this idea is meaningless, but the idea of “700+ level questions” is all over the place in the GMAT preparation marketplace, in the GMAT forums, etc. Why does something that is essentially meaningless get so much air time?

This also is complicated and underappreciated. In the days when I was a high school teacher, with my own classroom, I set the agenda. When a student asked about something that was clearly meaningless or clearly not productive, I could dismiss the question, explaining why we would not pursue that path. Everything was governed by the pedagogical principle that “*the teacher best understands the arc of the learning process*.”

In GMAT preparation, though, it’s an open market. Students are customers who choose one company over another, and in some ways, the marketing principle that “*The customer is always right*” can come into conflict with the pedagogical principle that “*the teacher best understands the arc of the learning process*.” If a student is convinced that “700+ questions” is a meaningful idea, and he wants to find a company that offers him such questions, then the answer I am giving in this blog article, while intellectually and pedagogically sound, might not be satisfying to him, and he may go elsewhere to find another company that offers him what he wants. Thus, in the GMAT preparation marketplace, individual companies have a high inducement to offer things for their marketing worth rather than for their pedagogical worth — hence, all the discussion of 700+ level questions. *Caveat emptor*! *Caveat discipulus*!

In the modern electronic marketspace, it can be hard for the student-customers to discern when a message they get is authentically beneficial, although perhaps not exactly what they were hoping to hear, vs. when a message exactly matches what they want to hear but is not necessarily in their best pedagogical interests. Of course, the very best indicator is ultimately not what the talking heads such as myself say, but the real results that students who use the product garner, and these are to be found in testimonials.

By all means, in your GMAT preparation, embrace a challenge. By all means, push yourself in mixed practice. Push yourself to develop new levels of understanding on each concept. Learn from your mistake, such that you never make the same mistake twice. Wrestle with the hardest questions you can find. Pursue excellence in every way. I simply dispute that you can reduce the pursuit of excellence to such a rigorously numerical system, such as knowing the “equivalent point value” of each question. The desire to quantify absolutely everything is one of the sicknesses of the modern world. Excellence ultimately transcends numbers. Excellence comes from the heart.

I realize that if you, as a student, have a burning question, receiving the response that your question is profoundly meaningless is not necessarily the most satisfying experience. I hope this blog gave you an appreciation for the complexity of this topic, as well as the nuances of marketing appeal vs. pedagogical integrity. If you have anything thoughts, questions, objections, we would love to hear them in the comments section below.

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