Consecutive Square Numbers

Transcript

Consecutive Square Numbers We start the chapter by introducing an algebraic identity. An identity is basically a mathematical "super-rule" that always works, no matter what numbers you use. The 3 Consecutive Squares Trick Our square numbers are 12, 22, 32, 42, 52, 62, … i.e. 1, 4, 9, 16, 25, 36… Now, Consider three consecutive square numbers Add the smallest and largest, then subtract twice the middle Result will always be 2 Example - Taking 1, 4, 9 Adding smallest and largest = 1 + 9 = 10 Twice middle = 2 * 4 = 8 Result = Smallest + Largest - 2 * Middle = 2 Example - Taking 9, 16, 25 Adding smallest and largest = 9 + 25 = 34 Twice middle = 2 * 16 = 32 Result = Smallest + Largest - 2 * Middle = 2 Algebra Proof Assuming three consecutive squares are (n – 1)2 , n2 , (n + 1) 2 Note: We take middle as n2 since it makes our calculation easier Doing our calculation (Smallest + Largest) – 2 × Middle = (n – 1)2 + (n + 1) 2 – 2 × n2 = (n2 + 1 – 2n) + (n2 + 1 + 2n) – 2 × n2 = (n2 + n2 – 2n2) + (2n – 2n) + (1 + 1) = 0 + 0 + 2 = 2