Overview
This lecture covers the concept of consecutive integers in arithmetic and algebra, focusing on how to represent consecutive, odd, and even integers using variables for problem-solving.
Consecutive Integers in Arithmetic
- Consecutive integers are numbers that follow one after another, e.g., 1, 2, 3.
- Consecutive even integers: numbers like 2, 4, 6, where each number is two more than the previous.
- Consecutive odd integers: numbers like 1, 3, 5, where each number is two more than the previous.
Consecutive Integers in Algebra
- Use a variable (commonly ( x )) to represent the first integer.
- The next consecutive integers are ( x + 1 ), ( x + 2 ), etc.
- Example: If ( x = 5 ), then the next two consecutive integers are 6 and 7.
Consecutive Odd Integers in Algebra
- Odd integers are represented by ( x ), ( x + 2 ), ( x + 4 ) where ( x ) is an odd number.
- Example: If ( x = 9 ), the next two consecutive odd integers are 11 and 13.
Consecutive Even Integers in Algebra
- Even integers are represented by ( x ), ( x + 2 ), ( x + 4 ) where ( x ) is an even number.
- Example: If ( x = 8 ), the next two consecutive even integers are 10 and 12.
Application & Patterns
- To model a sequence of consecutive (normal, odd, or even) integers in algebra, list as ( x, x + 1, x + 2 ) for regular, and ( x, x + 2, x + 4 ) for even/odd.
- For problem-solving, always define the variable for the first term, then each following term based on the pattern.
Key Terms & Definitions
- Consecutive Integers — Numbers that follow one another without gaps, differing by 1.
- Even Integers — Integers divisible by 2 (e.g., 2, 4, 6).
- Odd Integers — Integers not divisible by 2 (e.g., 1, 3, 5).
Action Items / Next Steps
- Practice writing algebraic expressions for sequences of consecutive, odd, and even integers.
- Prepare for application problems, such as finding the sum of consecutive integers.