Sequences

A sequence is an ordered list of numbers an−1, an, an+1, where n denotes where in the sequence the number is located. Sequence-based questions on the GMAT can look complicated, but once you learn to break down the notation, they’re not that tricky.

Example:

Before you dive into statement and answer evaluation, make sure you understand what you are looking for. Use Paraphrasing to state the information in the question in your own words. Then apply Critical Thinking to identify the pattern in the sequence you’re given. (Since this is the GMAT, there will always be a discernible pattern to the sequence even if it isn’t transparent immediately.)

Once you have a thorough understanding of the question stem and the sequence, look at the statements and answers. Picking numbers is a useful strategy if you’re not given concrete values for the terms in a sequence.

With a little patience and a strategic approach, you can attack even the most intimidating sequence questions effectively.

In-Format Question: Sequences on the GMAT

Now let’s use the Kaplan Method on a Problem Solving question dealing with sequences:

  1. In an increasing sequence of 8 consecutive even integers, the sum of the first 4 integers is 268. What is the sum of all the integers in the sequence?
    1. 552
    2. 568
    3. 574
    4. 586
    5. 590
Step 1: Analyze the Question

You are asked to determine the sum of all the integers in the sequence, and you are given the sum of the first 4 of 8 integers. As discussed in the Statistics chapter of this book, half of the numbers in a consecutive sequence with an even number of terms must be greater than and half must be less than the mean.

Step 2: State the Task

Since you are given the sum of the first half of the sequence, you can use the average formula to solve this strategically.

Step 3: Approach Strategically

Since the sum of 4 terms is 268, the average is 268 ÷ 4 = 67, which means the 4 even integers must be 64, 66, 68, and 70. This means that the next 4 must be 72, 74, 76, and 78. The sum of all 8 is 568, and that’s your answer. You could also apply Critical Thinking and realize that since the numbers in the sequence are consecutive even integers, the first term in the second group will be 8 larger than the first term in the first group, the second term in the second group 8 larger than the second term in the first group, and so on. This means that the final 4 numbers add up to 268 + (8 + 8 + 8 + 8), or 300. Then 268 + 300 = 568. Choice (B) is correct.

Step 4: Confirm Your Answer

The numbers in the answer choices here are not very spread out, so estimation is not likely to be very helpful. A quick double-check, especially using the alternative method above, will confirm that you didn’t make any mistakes.

Practice Set: Sequences on the GMAT

  1. l, m, n, o, p
    An arithmetic sequence is a sequence in which each term after the first is equal to the sum of the preceding term and a constant. If the list of letters shown above is an arithmetic sequence, which of the following must also be an arithmetic sequence?
    1. 3l, 3m, 3n, 3o, 3p
    2. l2, m2, n2, o2, p2
    3. l − 5, m − 5, n − 5, o − 5, p − 5
    1. I only
    2. II only
    3. III only
    4. I and III
    5. II and III
  2. In a certain sequence of positive integers, the term tn is given by the formula tn = 3(tn−2) + 2(tn−1) − 1 for all n ≥ 1. If t6 = 152 and t5 = 51, what is the value of t2?
    1. 1
    2. 2
    3. 3
    4. 6
    5. 17
  3. In the infinite sequence S, each term Sn after S2 is equal to the sum of the two terms Sn−1 and Sn−2. If S1 is 4, what is the value of S2?
    1. S3 = 7
    2. S4 = 10