Mathematics - Term 4 - Week 3

Patterns are sequences that follow a specific rule. For example, in the sequence 2, 4, 6, 8, the rule is to add 2 each time. Recognizing patterns helps us predict future numbers in the sequence. Patterns can be numerical, geometric, or algebraic. In algebra, we often express patterns using variables and equations, which allows us to generalize and solve problems efficiently.

Consider the pattern 5, 10, 15, 20. The rule here is to add 5. If we want to find the 10th term, we can use the formula for the nth term: a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference. Thus, a_10 = 5 + (10-1)5 = 5 + 45 = 50.

Let's work together on the sequence 3, 6, 9, 12. What is the rule? Students will identify that the rule is to add 3. Now, let's find the 5th term using the same formula: a_5 = 3 + (5-1)3. Students will calculate and share their answers. This encourages collaboration and reinforces the concept of patterns.

Now it's your turn! Solve the following problems independently: 1) Find the 7th term of the sequence 4, 8, 12, 16. 2) Write an algebraic expression for the pattern where each term increases by 7 starting from 1. Show your working clearly and be prepared to discuss your strategies with a partner.

• 8
• 9
• 10
• 11

Answer: 9

The pattern increases by 2 each time, so the next number after 7 is 9.

Answer: 2 + 4(n - 1)

This expression represents the nth term of the sequence.

• n = 10 + 10(n-1)
• n = 10n
• n = 20 + 10(n-1)
• n = 10n + 10

Answer: n = 10 + 10(n-1)

This equation correctly represents the pattern where each term increases by 10.

Answer: 17

The pattern increases by 3, so the 6th term is 2 + 3(6-1) = 17.

• 5
• 6
• 7
• 7

Answer: 7

The common difference is found by subtracting the first term from the second term: 14 - 7 = 7.

Answer: 10

The 4th term is calculated as 1 + 3(4-1) = 10.

• 2, 4, 8, 16
• 1, 3, 5, 7
• 10, 20, 30, 40
• 5, 10, 15, 25

Answer: 1, 3, 5, 7

An arithmetic sequence has a constant difference, which is 2 in this case.

Answer: Identify the first term and the common difference, then use the formula a_n = a_1 + (n-1)d.

This formula allows you to calculate any term in the sequence based on its position.