Occasionally, the SAT likes to challenge our minds with really diabolical questions. To make matters worse, knowing the mathematical concept required to solve a challenging question might not guarantee that you will be able to unravel the problem. A valuable yet regularly forgotten strategy is listing things out. Many students overlook this strategy, dismissing it as non-mathematical, brute force, or inelegant, but some questions can only be done by listing things out to reveal patterns.

The key is to translate the information on your scratch paper and use a sketch to visualize the problem. After visualizing the problem, we can then solve it.

Always remember that this is a timed test and you do not need to create anything fancy!

Consider the question below:

(Try to crack it before looking at the solution!)



All positive integers appear in the sequence above, each positive integer k appears in the sequence k times. In the sequence, each term after the first is greater than or equal to each of the terms before it. If the integer 12 first appears in the sequence as the nth term, what is the value of n?


Let’s start by visualizing the sequence based on the information available.

positive integer k appears in the sequence k times… Therefore, we have:

k=1,                             1

k=2,                             2, 2

k=3,                             3, 3, 3

k=4,                             4,4,4,4

Thus, the consequential sequence is:  1,2,2,3,3,3,4,4,4,4,… (Same as above)

From our visualization above, we know 1 appears once, 2 appears twice, 3 appears three times, etc.  All we need to do, then, is sum all the other term placements to get to 12:

 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66

The first 12 will appear after the first 66 terms:

 66 + 1 = 67

The value of n, then, is 67.

Note: The same strategy applies to Geometry problems that do not provide you with a sketch. Drawing out the shapes and angles always helps you get to the solution of the problem faster.


Join the conversation! 2 Comments

  1. Why did you not add all the numbers :
    For example 2(2)+3(3)

    • We did not add up the numbers because the question did not ask for the total sum of the first n integers, the question simply wanted to know where in the sequence the first 12 would appear. One wouldn’t see a 12 in the sequence until the 67th position.


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