Learn about Patterns through these example problems with detailed step-by-step solutions.
Example 1: Identifying a Number Pattern
Identify the pattern and find the next three numbers in the sequence: 2, 4, 6, 8, ...
Step 1: Look at the relationship between consecutive numbers in the sequence.
Step 2: Find the difference between the second number and the first number: $4 - 2 = 2$.
Step 3: Find the difference between the third number and the second number: $6 - 4 = 2$.
Step 4: Find the difference between the fourth number and the third number: $8 - 6 = 2$.
Step 5: The pattern is adding 2 to the previous number to get the next number. The rule is "Add 2".
Step 6: To find the next three numbers, continue adding 2 to the last number in the sequence (8).
Step 7: Next number 1: $8 + 2 = 10$.
Step 8: Next number 2: $10 + 2 = 12$.
Step 9: Next number 3: $12 + 2 = 14$.
Step 10: The next three numbers in the sequence are 10, 12, and 14.
Example 2: Identifying a Geometric Pattern
Look at the sequence of shapes: Circle, Square, Triangle, Circle, Square, Triangle, ... What is the next shape in the pattern?
● ■ ▲ ● ■ ▲ ...
Step 1: Observe the sequence of shapes and look for repetition or a rule.
Step 2: The sequence is Circle, Square, Triangle, Circle, Square, Triangle, ...
Step 3: Notice that the sequence of shapes "Circle, Square, Triangle" repeats.
Step 4: The pattern is a repeating unit of three shapes: Circle, Square, Triangle.
Step 5: The last shape shown is Triangle. After Triangle in the repeating unit comes Circle.
Step 6: Therefore, the next shape in the pattern is a Circle.
Example 3: Finding the Rule of a Pattern
What is the rule for the following number pattern: 5, 10, 15, 20, ...?
Step 1: Examine the relationship between consecutive numbers.
Step 2: Find the difference between the second and first number: $10 - 5 = 5$.
Step 3: Find the difference between the third and second number: $15 - 10 = 5$.
Step 4: Find the difference between the fourth and third number: $20 - 15 = 5$.
Step 5: The difference between consecutive numbers is consistently 5. This indicates a pattern of adding 5 each time.
Step 6: Alternatively, notice that each number is a multiple of 5: $5 \times 1 = 5$, $5 \times 2 = 10$, $5 \times 3 = 15$, $5 \times 4 = 20$.
Step 7: The rule for the pattern is "Add 5 to the previous number" or "Multiply the position number by 5".
