Binomial Theorem

A proof technique where we prove

A proof technique where we prove P(1) is true, then show that if P(k) is true, then P(k+1) is also true. This establishes the truth for all natural nu

(x + y)^n = ⁿC₀x^n +

(x + y)^n = ⁿC₀x^n + ⁿC₁x^(n-1)y + ⁿC₂x^(n-2)y² + ... + ⁿCₙy^n. Each term has form ⁿCᵣx^(n-r)y^r where r goes from 0 to n. The coefficients are binomi

The (r+1)th term in expansion

The (r+1)th term in expansion of (x+y)^n is Tᵣ₊₁ = ⁿCᵣx^(n-r)y^r. This formula allows finding any specific term without expanding the entire expressio

When n is even

When n is even: one middle term at position (n/2 + 1). When n is odd: two middle terms at positions (n+1)/2 and (n+3)/2. Example: For (x+y)⁸, middle t

For rational r and |x| <

For rational r and |x| < 1: (1+x)^r = 1 + rx + r(r-1)x²/2! + r(r-1)(r-2)x³/3! + ... This gives infinite terms unless r is a natural number. Condition