Posts tagged with: patterns

Strips of colored paper are made into loops and are chained to each other as shown above. The color pattern begins with the first loop and proceeds from left to right as red, blue, green, green, yellow, purple. This pattern is repeated for a chain of 96 loops. If the first few loops of the chain are removed so that the new beginning loop is no longer red and the new 16th loop is blue, what color will the new 51st loop in the chain be?

This is a classic example of a question writer getting carried away and taking things a step or two too far (not to say I am never guilty of this, of course).

First, let’s just take away all the rings before that blue one we know about. So let’s say the new new 1st ring is blue, making the old new 51st ring really the new new 36th ring. That also makes the new pattern we’re dealing with no longer start with red, but rather go thusly:

B, G, G, Y, P, R,
B, G, G, Y, P, R,

There are 6 colors in the pattern, so every 6th ring will be red. 36 is a multiple of 6, so the ring we’re asked about will be a red one.

(Read up more on repeating patterns.)

Pg 90 number 20 from Math Guide
After figuring out that Fantasia dates occur on multiples of 3 and Shayla dates on multiples of 6, is it an okay strategy to see that 279 is a multiple 3 and not 6 to get my answer choice? I’m a little confused about your solution.

Yep! That’s totally legit.

After the first term, each term in a sequence is k times the preceding term and p is the first term of the sequence. If the ratio of the seventh term to the third term is 16, what is a possible value of k?

Say the 3rd term is a. Then the 4th term is ka, the 5th term is (k^2)a, the 6th term is (k^3)a, and the 7th term is (k^4)a.

The question says that the ratio of the 7th term to the 3rd term, (k^4)a to a, is 16.

\dfrac{k^4 a}{a}=16


k = 2 or k = –2

Can you explain number 14 in diagnostic drill 3?
I tried solving it and i read the explanation of patterns.
i got the repeating units of the sequence which is 1,4,6,4,6,4,6
when he asked for the 53rd term should I divide 53 by 3 or 2?
I got confused because the one isn’t repeated again in the pattern.

Thanks in advance =)

For this one, the shortcut is to recognize that after the first term, every odd-numbered term is 6 and every even-numbered term is 4. The 53rd term is an odd-numbered term, so it’ll be 6.

The first term of a certain sequence is -2 and the second term is -4. Every term after the second term is obtained by dividing the sum of the previous two terms by 2. For example, the third term of the sequence is (-4)+ (-2)/2=-3. What is the sixth term of this sequence?

A) -3

B) -27/8

C) -21/8

D) -7/4

E) -1/8

List the terms:

  1. –4 (given)
  2. –2 (given)
  3. –3 (given, but also \frac{-4+(-2)}{2})
  4. –2.5 (\frac{-2+(-3)}{2})
  5. –2.75 (\frac{-3+(-2.5)}{2})
  6. –2.625 (\frac{-2.5+(-2.75)}{2})

Put that in your calculator and convert it to a fraction:

-2.625 = -\dfrac{21}{8}

Here’s a new quiz on patterns. For one week, it’ll be available to all site members, but then it’ll only be available to Math Guide owners. If you want the extra patterns practice, make sure you take this by 2/14 (Valentine’s Day)! How’s everyone else doing on this quiz?…

This content is for Math Guide owners only.
Log In Buy a Math Guide Verify your Math Guide ownership
broccoli fractal (source)

Pattern questions on the SAT aren’t super common, but they tend to give people all sorts of difficulty when they do appear. Let’s take one apart.

  1. A farmer is planting a row of plants. He first plants 2 broccoli plants, then 3 cabbage plants, then 1 apple tree, then 2 orange trees, then 1 dill weed plant. He repeats this pattern over and over again until he’s filled up all the land on his (very unorthodox) farm. What kind of plant is the 782nd one he plants?
    (A) cabbage
    (B) apple
    (C) broccoli
    (D) dill weed
    (E) orange

Obviously I’m taking some liberties with the writing style of the test makers, but take away the goofiness and this question could totally appear on your SAT. How do you solve it?

Start by writing a few iterations of the pattern on top of each other:

B, B, C, C, C, A, O, O, M,
B, B, C, C, C, A, O, O, M,
B, B, C, C, C, A, O, O, M, …

Now number the ends of the lines (in green here):

(1)B, B, C, C, C, A, O, O, M (9),

(10)B, B, C, C, C, A, O, O, M (18),

(18)B, B, C, C, C, A, O, O, M(27), …

Note that down the RIGHT side of your lines you’re basically counting up by multiples of 9. On the LEFT side, you’re also counting up by 9’s, but in a slightly more confusing way. The mistake a lot of people make on pattern questions is that they start counting from the beginning of the pattern, and they get all screwed up. Count from the end of the pattern to make this easy on yourself!

Every even multiple of 9 will be a dill weed plant. We want the 782nd plant. 782 ÷ 9 = 86 remainder 8, which means the 782nd plant will be the 8th in our pattern. In other words, it’ll be an orange tree. (E) is our answer. Another way to think of this: the 783rd plant will be a dill weed plant, so the 782nd will be an orange tree.

No sweat, right? Sit down, SAT. You can’t slow me down with that weak sauce.

Of course, repeating patterns like the one above are only one kind of pattern that might get thrown at you, just the kind that seem to give students the most trouble, in my experience. The drill below contains more pattern-type questions.

Try these examples:

You need to be registered and logged in to take this quiz. Log in or Register

1, 7, 49, 343, …
  1. Each term in the sequence above after the first one is determined by multiplying the previous term by 7. What will be the units (ones) digit of the 96th term?
    (A) 9
    (B) 7
    (C) 5
    (D) 3
    (E) 1

Answer and explanation below:

Usually, when the SAT asks you to find something like the 96th term, they’re really asking you to recognize the pattern that exists in all the terms. Fill the sequence in a little more and the pattern should reveal itself:

1, 7, 49, 343, 2401, 16807, 117649,…

See it? The units digits follow a very simple pattern: 1, 7, 9, 3, 1, 7, 9, 3, etc.

So now all you need to do is figure out the 96th term. Every 4th term is 3, and 96 is evenly divisible by 4, so the 96th term has to be 3!

(D) is your answer.