Find how many numbers lie between two consecutive perfect squares. Do you notice a pattern?
Given: Consecutive perfect squares n² and (n+1)²
Formula: Number of integers between them = (n+1)² - n² - 1
Calculation: (n+1)² - n² - 1 = n² + 2n + 1 - n² - 1 = 2n
Examples:
- Between 1² = 1 and 2² = 4: 2 numbers (2, 3)
- Between 2² = 4 and 3² = 9: 4 numbers (5, 6, 7, 8)
Answer: Between n² and (n+1)², there are exactly 2n numbers.
Pattern: The count equals 2n where n is the smaller of the two consecutive numbers being squared.
Explanation
The chapter explores the relationship between consecutive squares. The formula (n+1)² - n² - 1 = 2n gives the count of integers strictly between n² and (n+1)².
Solution Steps
Step 1: Consider consecutive squares n² and (n+1)²
Step 2: Count integers between them: (n+1)² - n² - 1
Step 3: Simplify: n² + 2n + 1 - n² - 1 = 2n
Step 4: The pattern: between n² and (n+1)², there are 2n integers