Identifying Pattern Relationships

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 the 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.