The variable n represents the figure number in the following algebraic pattern rules:
a) 4n b) 2n + 2 c) 3n + 1 d d) 4n - 1
Which of these pattern rules describes the toothpick pattern shown below? Explain how you know
To determine which pattern rule describes the toothpick pattern, we need to analyze the given options and understand how they relate to the figure number, represented by the variable n.
a) 4n: This pattern rule suggests that the number of toothpicks is calculated by multiplying the figure number n by 4. For example, if n = 1, then the number of toothpicks would be 4 * 1 = 4. However, we cannot definitively determine if this is correct without analyzing the actual toothpick pattern.
b) 2n + 2: According to this pattern rule, the number of toothpicks is given by doubling the figure number n and then adding 2. For instance, if n = 1, then the number of toothpicks would be (2 * 1) + 2 = 4.
c) 3n + 1: This pattern rule states that the number of toothpicks is obtained by multiplying the figure number n by 3 and then adding 1. For instance, if n = 1, then the number of toothpicks would be (3 * 1) + 1 = 4.
d) 4n - 1: According to this pattern rule, the number of toothpicks is calculated by multiplying the figure number n by 4 and then subtracting 1. For example, if n = 1, then the number of toothpicks would be (4 * 1) - 1 = 3.
Now, let's analyze the toothpick pattern shown below:
1 4 9 16
1 8 18 32
1 12 27 48
From the given toothpick pattern, we can see that the number of toothpicks in each figure increases quadratically.
To verify the correct pattern rule, we can pick any figure and calculate the number of toothpicks based on each pattern rule option. Let's take the second figure.
According to option a) 4n: 4 * 2 = 8 (incorrect)
Option b) 2n + 2: (2 * 2) + 2 = 6 (incorrect)
Option c) 3n + 1: (3 * 2) + 1 = 7 (incorrect)
Option d) 4n - 1: (4 * 2) - 1 = 7 (incorrect)
Since none of the pattern rule options perfectly match the toothpick pattern, we can conclude that none of the given options accurately describe the toothpick pattern shown.