First of all, how can you tell if the question you’re working on is a pattern problem? Sometimes it’s obvious:
Q. Gisela is stringing beads by following a pattern..
When a sequence of numbers is presented before the question begins, there’s usually a pattern:
4, 7, 10, . . .
Q. In the sequence above…
You can also follow this rule: If you can solve a question by doing the same thing more than 5 times, there has to be a pattern. The SAT is a reasoning test; the test writers don’t want you to do the same thing repeatedly.
Q. What is the color of the 73rd light?
You don’t need to see any more to know that this is a pattern problem.
So how do you solve them? On easier questions, you can often just write out the next few values until you get the answer:
3, 8, 18, 38
Q. In the sequence above, each term after the first is obtained by doubling the previous number and adding 2. What is the sixth term in the sequence?
2(38) + 2 = 78 (5th term), 2(78) + 2 = 158.
On harder questions, it’s too time consuming to write out all the values. The key rule is that the last element of a pattern repeats on its multiple. I know that’s not plain English, so make sure you understand it before you continue with this lesson. If red, green, orange is repeated over and over, orange is the last element in a pattern of 3 things, so orange will appear at every multiple of 3. That is, the 3rd, 6th, 9th etc. thing will be orange. If the numbers 5, 3, 2, 7 are repeated, 7 will be the 4th, 8th, 12th etc. number (note that the value of the last number doesn’t matter; it’s position does).
In the number depicted above, the digits 7, 6, 5, 4, and 3 repeat indefinitely to the right of the decimal point. What is the 73rd digit to the right of the decimal point?
There are 5 digits in the pattern, so the last one (3) will repeat on every multiple of 5. Just pick one that’s close to the 73rd – the 70th will be a 3, the 71st will be 7, the 72nd will be 6, and the 73rd will be 5.
You’ll have to judge whether it’s worth trying to write out all the values. If you have to write out more than 25 steps, don’t even consider that approach, even if the steps are easy.
In some pattern questions, you have to figure out what the pattern is for yourself.
Q. On a square game board with m rows of m squares each, p of the squares lie on the edge of the board. Which of the following could be the value of p?
One way to solve this is to pick a board of a reasonable size and see what happens. If you pick a standard checkerboard, which is 8 X 8, you can see that 32 is an incorrect answer. True, there are 8 squares along each edge, but you don’t want to count the corner squares twice. The correct answer is 32 – 4 = 28. So the answer will be 4p – 4. Now you can just try some values of p until you get it.
If that confuses you, another way is to start will the smallest possible board (2X2), which has all 4 squares on the edge. A 3X3 board will have 8 squares on the edge, and a 4X4 board will have 12 squares on the edge. Aha – the answer will be a multiple of 4!
Finally, the test writers try to trick you into picking an answer that’s off by 1 place.
Q. Jim has a colony of amoebas that double their number every 4 hours. If he starts with 8 amoebas, how many does he have after 20 hours?
Careful, he starts with 8 amoebas – that’s when zero hours have passed. So after 4 hours, he has 16 amoebas (8 hrs – 32 am, 12 hrs – 64 am, 16 hrs – 128 am, 20 hrs – 256 amoebas).
6, 17, 28, . . .
Q. In the sequence above, each term after the first is found by adding 11 to the preceding term. Which of the following terms is equal to 6 + 33(11)?
The first term is 6. The 2nd term is 6 + 1(11), the 3rd term is 6 + 2(11), so the 34th will be 6 + 33(11).
8, 41, 206, . . .
Q. In the sequence above, each term after the first is obtained by multiplying the previous number by 5 and adding 1. What is the last digit of the 98th term in the sequence?
Write out a few more: 8, 41, 206, 1031, 5136. The 2nd, 4th, and all even terms will end in 1.
I have found that there is an average of about 1 pattern question per SAT.