Identifying Patterns
A mathematical pattern is an object or group of objects that possesses regularity or repetition (which could be visual, arithmetic, etc.). For example, \( 1, 2, 4, 8, 16, 32, …\) is a pattern made of numbers (called a sequence), and this pattern is characterized by doubling (i.e., each term is twice as large as the term before).
Finding and describing patterns is at the heart of mathematics. While sometimes patterns can lead us astray (for example, the Greeks believed false things about perfect numbers because of patterns that didn't continue), the ability to recognize and extend patterns is extremely important.
Describing Visual Patterns
Main Article: recognizing visual patterns
Searching for visual patterns can be as simple as identifying the change from one item in a sequence to the next. Some possible changes to look for include
- changes in color
- rotation
- vertical or horizontal translation
- changes in shape
- changes in size.
When looking for visual patterns, it is a good practice to make a hypothesis based on one or two terms and then test it against an additional item to see if your expected pattern matches the entire sequence.
Being able to recognize visual patterns will allow you to solve problems like this:
What comes next?
image
If we track the blue square, we see that it goes from the top, down to the second, and then to the third, so it will likely be in the fourth (at the bottom). Notice that there are \(2\) options with the blue square at the bottom.If we look at any other square, say the yellow square, we see that it similarly moves downwards, and that after it is the fourth, it moves to the top. Hence, the \(2^{\text{nd}}\) square should be yellow, the first square should be green, and the third square should be red.
Thus, the answer is B. \( _ \square \)
Describing Numerical Patterns
In order to determine the set of manipulations that will produce a specific pattern, it is necessary to compare the sequence provided with the results obtained by using the proposed generating rule. When looking for an expression that describes a pattern, it is important to check every term to make sure your conjecture fits all the evidence.
Some patterns might appear to match a certain rule, but then diverge after the initial terms. For example, in the sequence \( 3, 5, 7, 11, 13, 17, \dots \), someone analyzing only the first three numbers might think the pattern includes all odd numbers, but further inspection reveals that \(9\) is missing, and the series is actually primes.
Which of the following rules describes the sequence \( 2, 4, 10, 28, 82, \dots \)?
A) Square the previous term.
B) Multiply the previous term by \(3\) and subtract \(2.\)
C) Multiply the previous term by \(2\) and add \(2.\)
D) Cube the previous term and subtract \(4.\)
Looking at the first two terms, \(2\) and \(4,\) we see that rules A, B, and D work, but rule C does not:
\[ 2 \times 2 + 2 = 6 \not = 4. \]
Looking at the next two terms, \(4\) and \(10,\) we see that rule A fails: \( 4^2 = 16 \not = 10\), and rule D also fails:
\[ 4^3 - 4 = 60 \not = 10. \]
Rule B works for every pair of terms in the sequence. Thus the answer is B. \(_\square\)
Which of the following rules describes the sequence \( 3, 4, 6, 8, 12, \dots \)?
A) Composite numbers
B) Adding \(1\) to prime numbers
C) Subtracting \(1\) from squares
D) Prime numbers
First, seeing the first two numbers, we can easily say that A and D couldn't satisfy the sequence. (As \(3\) is a prime and \(4\) is a composite.)
Let's take the statement C: as \(1\) is the first square, subtracting \(1\) to it gives \(0.\) But, the first number is \(3,\) so statement C is False.
Since
\[\begin{array} &2+1 = 3 , &3+ 1 = 4 , &5+ 1 = 6 , &7+ 1 = 8 , &11+ 1 = 12, \end{array}\]
the correct answer is B. \(_\square\)
How many cubes are in the fifth step of this pattern?
The pattern is presented visually and the question asks for the next term. This problem could be solved by inspecting the pattern and drawing the next term, or by writing an algebraic equation.
Inspecting the image reveals that, for each term of the pattern, one cube is added to the horizontal line and one cube is added to the vertical stack. The pattern for the number of cubes is \(1,3,5,7,...\)
The 5th drawing will have a total of 9 cubes.
While it's simple to draw the pattern for the fifth item in this sequence, the task gets more challenging as the number of boxes increases. Drawing the pattern is an inefficient way to find, for example, the 43rd term in the series.
What algebraic identity is this?