Numeric Patterns and Sequences

Numeric Patterns Introduction

• Numeric patterns are number patterns.
• Geometric patterns involve diagrams.
• Geometric patterns use diagrams to present number patterns.
• Numeric patterns are sequences of numbers that follow a specific rule.

Understanding Sequences

• Sequences can involve addition, subtraction, multiplication, or division.
• Identify the rule by determining the operation between consecutive numbers.
• Consider what operation transforms one number to the next in the sequence to find the rule.
• Constant difference and forming a number pattern are the 2 main focuses when working with numeric sequences.

Constant Difference

• Constant difference refers to the consistent difference between consecutive numbers in a pattern.
• The constant difference (d) is found by subtracting a term from its subsequent term (e.g., t2 - t1).
• d = t2 - t1 = t3 - t2, etc.
• Moving forward in the sequence means the constant difference is added to obtain the next term.
• Adding the constant difference to the previous term yields the next term.

General Rule

• Finding the next terms and calculating the general rule on a number pattern with a constant difference are different mathematical tasks.
• The general rule expresses the relationship between the input (term number) and the output (term value).
• Input-output concept: the rule transforms the input (term number) into the output (term value).
• The general rule uses 'n', representing the term number.
• General rule formula: tn = (constant difference) * n + (number to be added/subtracted).
• The number to be added/subtracted ensures the rule yields the correct term value for each position.

General Rule Calculation

• The constant difference is represented by "d", and is a known value.

General Rule Calculation (continued)

• Find the number to be added/subtracted by determining what must be added/subtracted after multiplying the constant difference by 'n' to get the term in its position.
• For term position 1 (n=1), consider an example where 2 * 1 = 2, implies adding 3 to get 5 (the first position).
• Verify the found number with another position (n=2) to validate the constant in the rule.
• General rule example: tn = 2n + 3.
• Once the general rule is determined, any term at any position can be calculated.
• To find any term, substitute the desired position (n) into the general rule.

Constant Difference Example

• Given sequence: 6, 10, 14, 18,...
• First term (t1) = 6
• Second term (t2) = 10
• Third term (t3) = 14
• Fourth term (t4) = 18

Stating the Rule in Words

• Describe the operations required to obtain subsequent terms.
• Start with 6, then add 4 to get 10, repeating the addition of 4 continuously to find the next terms.

Constant Difference (Example continued)

• Constant difference = 4, found by subtracting consecutive terms.
• Next three terms: 22, 26, 30, obtained by adding 4 to the previous term.

Determining General Rule (Example continued)

• Use formula: tn = (constant difference) * n + (number to be added).
• Constant difference = 4 so tn = 4n + (number to be added).
• To determine what must be added to get the first term: solving 4 * 1 + x = 6 gives x = 2.
• Validating with another position where n is 2: solving 4 * 2 + x = 10 gives x = 2.
• General rule: tn = 4n + 2.

Calculating the nth term

• For n = 10, t10 = 4(10) + 2 = 42, so the 10th term of the sequence is 42.
• For n = 100, t100 = 4(100) + 2 = 402, so the 100th term of the sequence is 402.

Finding Term Position

• Determining the term position in the sequence involves finding the 'n' (position) when the term value (tn) is known.
• If the term is 242, find n when tn = 242, answering the question: 242 is at which position in the sequence?
• Equate the general rule with the given term value: 4n + 2 = 242, then solve for n.
• Solving 4n + 2 = 242 gives n = 60.
• If tn = 242 and n = 60 then t60 = 242.