# SAT Math Strategies: Making Lists to Find Patterns

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!)

**Question**:

1,2,2,3,3,3,4,4,4,4,…

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 *n*th term, what is the value of *n*?

**Solution**:

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.

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.