Question

1 exploration: describing a pattern (continued)

c. n = 1 n = 2 n = 3 n = 4 n = 5

number of rows, n 1 2 3 4 5
number of dots, y

communicate your answer

1. how can you use an arithmetic sequence to describe a pattern? give an example from real life.
2. in chemistry, water is called h₂o because each molecule of water has two hydrogen atoms and one oxygen atom. describe the pattern shown below. use the pattern to determine the number of atoms in 23 molecules.

n = 1 n = 2 n = 3 n = 4 n = 5

Explanation:

1. Filling the table for dot - pattern

Step1: Count dots for \(n = 1\)

For \(n=1\), there are 2 dots. So when \(n = 1\), \(y=2\).

Step2: Count dots for \(n = 2\)

For \(n = 2\), there are 4 dots. So when \(n = 2\), \(y = 4\).

Step3: Count dots for \(n = 3\)

For \(n=3\), there are 6 dots. So when \(n = 3\), \(y = 6\).

Step4: Count dots for \(n = 4\)

For \(n = 4\), there are 8 dots. So when \(n = 4\), \(y = 8\).

Step5: Count dots for \(n = 5\)

For \(n=5\), there are 10 dots. So when \(n = 5\), \(y = 10\).

An arithmetic sequence has a common difference between consecutive terms. For example, in the cost of buying stamps. If one - stamp costs $0.50, the cost of 2 stamps is $1.00, 3 stamps is $1.50. The sequence of costs LXI0 has a common difference of $0.50. The general formula for an arithmetic sequence is \(a_n=a_1+(n - 1)d\), where \(a_1\) is the first - term and \(d\) is the common difference.

Step1: Determine number of atoms in one molecule

One molecule of water (\(H_2O\)) has \(2 + 1=3\) atoms.

Step2: Find general formula for number of atoms in \(n\) molecules

The number of atoms in \(n\) molecules of water follows a pattern where the number of atoms \(A = 3n\).

Step3: Calculate number of atoms in 23 molecules

Substitute \(n = 23\) into the formula \(A=3n\). So \(A=3\times23 = 69\).

Answer:

Number of rows, \(n\)	1	2	3	4	5

2. Using arithmetic sequence to describe a pattern