Sequence questions come up fairly regularly on the GMAT quantitative section. One of the biggest problems students report on these questions is that they can’t determine what the terms in sequence should actually be. As such, the first important thing to determine is the value of the first few elements of the sequence. Without this information, the question seems much more abstract and difficult to follow.
What’s important to note is that any sequence is predicated on specific rules. To take a famous example, the Fibonacci sequence is defined as a1 = 1 and a2 = 1, and then for all subsequent terms: an = an-1 + an-2. Breaking through the math, the third term will be the sum of the first and second. The fourth term will be the sum of the second and third, etc. Turning the general an formula into a1 = 1, a2 = 1, a3 = 2, a4 = 3, a5 = 5, a6 = 8, a7 = 13… makes it a lot easier to grasp what is happening in this sequence.
Of course, simply determining the first few elements of a sequence is never sufficient to solve the problem. It is, however, a necessary step towards understanding how to answer the question. Knowing what the sequence looks like is important, because knowing is half the battle (G.I. Joe). There are still potentially other pitfalls that must be avoided, but having the rules of the sequence clearly understood helps avoid some of the clever pitfalls the test makers use to make questions more difficult.
Let’s look at a data sufficiency sequence question that highlights these issues:
The infinite sequence a1, a2, … an, … is such that a1 = x, a2 = y, a3 = z, a4 = 3 and an = an-4 for n > 4. What is the sum of the first 98 terms of the sequence?
(1) x = 5
(2) y + z = 2
(A) Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
(B) Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
(C) Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.
(D) Each statement alone is sufficient to answer the question.
(E) Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.
Before even looking at the two statements, let’s try and understand what the sequence is telling us about itself. It’s an infinite sequence where the first three terms are the variables x, y and z, and the fourth term is 3. After the fourth term, the numbers simply repeat in the same pattern. So the sequence looks like x, y, z, 3, x, y, z, 3, x, y, z, 3 etc. This helps us figure out what the question is actually asking, which in this case is a sum involving 3 separate variables (x, y and z) and only two statements. (looks like E at this preliminary stage!)
Statement 1 gives us a precise value of x. So basically I now need to know the sum of 5 + 3 + y + z. I still don’t have any value for y or z, so I can’t find an actual value for this sum. Statement 1 will be insufficient because I still have two unknowns.
Statement 2 on its own gives us values of y and z, but only as a sum. Without a value of x, this is still insufficient as the sum of the first four numbers will be x + 2 + 3. Statement 2 will be insufficient, so the answer will be either C or E.
Combining the statements, I have values for x and y + z, and thus if the question is asking x + y + z + 3, I know this must end up being 5 + 2 + 3 = 10. I know with 100% certainty that the sum of the first four terms will be exactly 10. The one caveat to be aware of is that we don’t have values for y and z, only for y + z. So y and z could be 0.5 and 1.5 or they could both be 1 (or -100 and +102) and we’d never know the difference.
This issue may be important to answer the question, as we are being asked for a sum of a number of elements. If they wanted to know the sum of the first element, statement 1 lets us know that it must be 5. If they wanted to know the sum of the first three elements, both statements together confirm that it must be 7. However, if the question was about the sum of the first two elements, then the answer could be 6 or 5.1 or even -95. We cannot determine the sum of the first two numbers with precision. And since this pattern repeats every 4 numbers, we cannot determine the sum of the first six elements, or the first ten elements, etc.
This question in particular is asking for the sum of the first 98 elements, so we must determine whether this is one of the sums that separates y and z. If it does, then we don’t know the exact sum. If it doesn’t, then we have sufficient data to determine the exact sum. The pattern repeats every 4 numbers, so every multiple of 4 will add 10 to the sum. We can use multiples of 4 to quickly determine that the first 40 or the first 80 are easy to calculate. After that, you can just add bounds in 4 to go from 80 to 84 to 88 to 92 to 96. Adding two more numbers would mean adding x and y again, which is the one spot we wanted to avoid. The answer to this question is thus E as we cannot determine the value of y with any certainty whatsoever. Answer choice E is correct in this case.
Had this question been the sum of the first 97 elements, we could have calculated it with certainty (10 x 24 + 5 or 245). Had this question been the sum of the first 99 elements, we could have also calculated it with certainty (10 x 24 + 7 or 247). The sum of this sequence is unclear if the remainder of the division by four is two (same concept as modulo, which isn’t explicitly tested on the GMAT but is nonetheless good to know). On sequence questions, determining the first few elements helps concretize the concept and make the numbers easier to understand. Once you do that, you’ll see your accuracy rate climb as a direct consequence (i.e. con-sequence!).
Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam. After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.