Binomial Theorem (Positive Integer Index)
(x + a)n = nC0 xn + nC1 xn-1a + nC2 xn-2a2 + ... + nCn an
Key Facts
- Total terms = n + 1
- Sum of powers of x and a in each term = n
- Binomial coefficients: nCr = n! / (r!(n−r)!)
- nCx = nCy ⇒ x = y or x + y = n
Greatest Binomial Coefficient
n even → nCn/2 is maximum
n odd → nC(n−1)/2 = nC(n+1)/2 (both maximum)
General Term & Middle Term
Tr+1 = nCr xn−r ar
Middle Term(s)
n even → One middle term: T(n/2)+1 = nCn/2 xn/2 an/2
n odd → Two middle terms: T((n+1)/2) and T((n+3)/2)
Ratio of Consecutive Terms
Tr+1 / Tr = (n−r+1)/r × (a/x)
Important Identities & Sums
- Sum of coefficients: (1+1)n = 2nC0 + nC1 + ... + nCn = 2n
- n × 2n−1 = nC1 + 2nC2 + ... + nnCn
- n+1C1 + n+1C2 + ... + n+1Cn+1 = 2n+1 − 1
- Alternating sum: (1−1)n = 0 = 0 = nC0 − nC1 + nC2 − ...
Numerically Greatest Term
|Tr+1| / |Tr| = (n−r+1)/r × |a/x| ≥ 1
⇒ r ≤ (n+1)|a/x| / (1 + |a/x|)
⇒ r ≤ (n+1)|a/x| / (1 + |a/x|)
Let r₀ = greatest integer ≤ (n+1)|a/x| / (1 + |a/x|)
Then Tr₀+1 is numerically greatest.
If the value is integer, then Tr₀ = Tr₀+1 (both greatest)
Binomial Theorem for Any Index
(1 + x)n = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + ... ∞
Valid when |x| < 1 (n any real number)
Valid when |x| < 1 (n any real number)
Important Expansions (|x| < 1)
- (1 + x)−1 = 1 − x + x² − x³ + x⁴ − ... ∞
- (1 − x)−1 = 1 + x + x² + x³ + x⁴ + ... ∞
- (1 + x)−2 = 1 − 2x + 3x² − 4x³ + ...
JEE Main Tips & Weightage
One of the most important & scoring chapters — 4–6 questions every year!
Most Important Topics
- Finding general term Tr+1
- Middle term(s)
- Numerically greatest term
- Finding coefficient of xr
- Expansions with |x| < 1 (infinite series)
- Sum of series using binomial
Quick Revision Formulae
- Total terms = n+1
- Greatest coeff → floor or ceil of (n+1)/2
- Greatest term → r ≈ (n+1)|a/x| / (1 + |a/x|)
- Sum of coeffs = 2ⁿ
- Alternating sum = 0 (n > 0)