Binomial expansion — A/L Combined Mathematics

The Binomial Theorem for a positive integer index n ∈ ℤ⁺ states that: (a + b)ⁿ = ∑_r=0ⁿ ⁿC_r a^n-r b^r = ⁿC₀ aⁿ + ⁿC₁ a^n-1 b + ... + ⁿC_n bⁿ, where the binomial coefficient ⁿC_r = n! / [r! (n - r)!]. Essential properties include: (1) Total terms in the expansion is exactly (n + 1); (2) The sum of powers of a and b in every term equals n; (3) Symmetry of coefficients: ⁿC_r = ⁿC_n-r; (4) Pascal's Identity: ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r; (5) Sum of all coefficients: setting a = b = 1 gives ∑_r=0ⁿ ⁿC_r = 2ⁿ; (6) Alternating sum: setting a = 1, b = -1 gives ∑_r=0ⁿ (-1)^r ⁿC_r = 0.

Check before you answer (A/L Pure Maths Paper 1)

1. General Term Identification: Writing the (r + 1)-th general term accurately as T_r+1 = ⁿC_r a^n-r b^r including negative signs from b.
2. Coefficient Identities: Proving summation identities like ⁿC₁ + 2ⁿC₂ + ... + nⁿC_n = n 2^n-1 using differentiation or algebraic expansion.
3. Pascal's Triangle Link: Linking binomial coefficients to row entries of Pascal's triangle and applying ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r.