Binomial Theorem

Statement of Binomial Theorem

• (x + a)ⁿ = ⁿC₀xⁿ + ⁿC₁xⁿ⁻¹a + ⁿC₂xⁿ⁻²a² + … + ⁿC_naⁿ, where n ∈ N
• Binomial coefficients: ⁿC_r = n! / [r!(n−r)!]
• General term: T_r+1 = ⁿC_r x^n−r a^r
• Total number of terms = n + 1
• Sum of powers of x and a in each term = n
• Equidistant binomial coefficients from start and end are equal

Greatest Binomial Coefficient

• If n is even: Greatest coefficient is ⁿC_n/2
• If n is odd: Greatest coefficients are ⁿC_(n−1)/2 = ⁿC_(n+1)/2

Middle Term(s) of the Expansion

• If n is even: Middle term is T_(n/2)+1
• If n is odd: Middle terms are T_(n+1)/2 and T_(n+3)/2

To Determine a Particular Term

• In the expansion of (x + 1/x)ⁿ, if x^m occurs in T_r+1, then r = (n − m)/2
• The term independent of x occurs when r = n/2 (n must be even)

Important Binomial Coefficient Properties

• ⁿC₀ + ⁿC₁ + … + ⁿC_n = 2ⁿ
• ⁿC₀ − ⁿC₁ + ⁿC₂ − … + (−1)ⁿⁿC_n = 0
• ⁿC₀ + ⁿC₂ + ⁿC₄ + … = 2ⁿ⁻¹
• ⁿC₁ + ⁿC₃ + ⁿC₅ + … = 2ⁿ⁻¹
• ⁿC₁ + 2ⁿC₂ + 3ⁿC₃ + … + nⁿC_n = n·2ⁿ⁻¹

Numerically Greatest Term

• For expansion of (a + x)ⁿ:
• Greatest term is T_r+1, where r = ⌊(n + 1) / (1 + |a/x|)⌋
• If (n + 1)/(1 + |a/x|) is a natural number, then both T_r and T_r+1 are greatest

Binomial Theorem for Any Index

• If n ∈ Q and |x| < 1:
• (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + …
• Number of terms is infinite

Standard Expansions (|x| < 1)

• (1 − x)⁻¹ = 1 + x + x² + x³ + …
• (1 + x)⁻¹ = 1 − x + x² − x³ + …
• (1 − x)⁻² = 1 + 2x + 3x² + 4x³ + …
• (1 + x)⁻² = 1 − 2x + 3x² − 4x³ + …
• (1 − x)⁻³ = 1 + 3x + 6x² + 10x³ + …

Some Important Results

• If coefficients of r^th, (r+1)^th, (r+2)^th terms of (1 + x)ⁿ are in H.P., then n + (n − 2r)² = 0
• If coefficients of r^th, (r+1)^th, (r+2)^th terms are in A.P., then n² − n(4r + 1) + 4r² − 2 = 0
• Number of terms in expansion of (x₁ + x₂ + … + x_r)ⁿ is ^n+r−1C_r−1