Overview
This lecture introduces constant difference patterns, also known as arithmetic sequences, explaining how to recognize them, write their formulas, and solve related problems.
Introduction to Constant Difference Patterns
- A constant difference pattern is a sequence where the same number is added or subtracted each time.
- Example: In 5, 8, 11, 14..., the constant difference is +3.
- Example: In 20, 18, 16, 14..., the constant difference is -2.
- Such patterns are commonly called arithmetic sequences.
Identifying the Pattern and the Rule
- To describe a pattern in words, state the operation: "Add 3" or "Subtract 2".
- To write a rule in algebra, use: ( t_n = d \times n + c ), where ( d ) is the difference and ( c ) is the starting number.
- To find ( c ), imagine or calculate what comes before the first term.
Writing and Using the Formula
- Given the sequence, identify the constant difference and the first term to form the rule.
- Example: For 10, 14, 18..., difference is +4, so rule: ( t_n = 4n + 6 ).
- For decreasing sequences, the rule uses a negative difference (e.g., ( t_n = -2n + 22 )).
Calculating Terms and Values
- Each position in the sequence is called a "term" (term 1, term 2, etc.).
- To find a specific term's value, plug the term number into ( n ) in the formula.
- Example: With ( t_n = 3n + 17 ), term 42 is ( 3 \times 42 + 17 = 143 ).
- To find which term has a specific value, set the formula equal to that value and solve for ( n ).
Solving for Unknown Term Numbers
- Rearrange ( t_n = d n + c ) to solve for ( n ): ( n = \frac{(value - c)}{d} ).
- Example: For ( t_n = 5n + 6 ), if value is 261, then ( n = \frac{261-6}{5} = 51 ).
Practice with Tables and Patterns
- Tables may list terms and values; use the pattern to find missing terms.
- Continue adding/subtracting the difference for successive terms to fill in tables.
Key Terms & Definitions
- Constant Difference/Arithmetic Sequence — a pattern where the same amount is added/subtracted each step.
- Term — the position number in the sequence.
- Value — the number found at a specific term.
- Rule/Formula — an algebraic expression to find any term’s value, usually ( t_n = d n + c ).
Action Items / Next Steps
- Practice identifying patterns, writing formulas, and solving for specific terms or positions.
- Complete table exercises by finding missing sequence values for given term numbers.