Binomial Theorem

Expansion of (a+b)^n, general term, middle term, properties of binomial coefficients.

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Expansion and general term

(a+b)^n, T_(r+1), middle term.

Binomial theorem — general term, middle term, special coefficients

Notes

For any positive integer n:

(a + b)ⁿ = Σ_{r=0}^{n} C(n, r) · a^(n−r) · b^r

= C(n,0) aⁿ + C(n,1) a^(n−1) b + C(n,2) a^(n−2) b² + ... + C(n,n) bⁿ

where C(n, r) = n! / (r!(n−r)!) is the binomial coefficient.

General term: T_(r+1) = C(n, r) · a^(n−r) · b^r

This is the (r+1)-th term, useful for finding any specific term.

MIDDLE TERM

If n is even: middle term is T_(n/2 + 1) — exactly one middle term.
If n is odd: two middle terms T_((n+1)/2) and T_((n+3)/2).

Worked example. Find the middle term in (x + 2)¹⁰.

n=10 (even). Middle term = T_6 = C(10, 5) · x⁵ · 2⁵ = 252 · x⁵ · 32 = 8064 x⁵.

Term independent of x: the term where powers of x cancel out. Set the x-exponent in T_(r+1) to zero and solve for r.

Worked example. Find the term independent of x in (x − 2/x²)⁹.

T_(r+1) = C(9, r) · x^(9−r) · (−2/x²)^r = C(9, r) · (−2)^r · x^(9 − r − 2r) = C(9, r) · (−2)^r · x^(9 − 3r).

For x^0: 9 − 3r = 0 → r = 3.

T_4 = C(9, 3) · (−2)³ = 84 · (−8) = −672.

Coefficient of x^k: use the general term, set x-power = k, solve for r, plug back.

PROPERTIES OF BINOMIAL COEFFICIENTS

Sum of all coefficients: put a = b = 1 in (a + b)ⁿ:
C(n, 0) + C(n, 1) + C(n, 2) + ... + C(n, n) = 2ⁿ

Alternating sum: put a = 1, b = −1:
C(n, 0) − C(n, 1) + C(n, 2) − ... = 0

Sum of even-position coefficients = sum of odd-position = 2^(n−1) (from combining the above two).

Pascal's identity:
C(n, r) + C(n, r−1) = C(n+1, r)

Symmetry: C(n, r) = C(n, n−r).

Vandermonde's identity:
C(m+n, k) = Σ_r C(m, r) C(n, k−r)

Greatest coefficient and greatest term:

If n is even: greatest C(n, r) is at r = n/2.
If n is odd: greatest C(n, r) is at r = (n−1)/2 or r = (n+1)/2 (both equal).

For greatest term in (a + b)ⁿ, the ratio T_(r+1)/T_r ≥ 1 gives the range of r where terms are still growing. Take the largest such r.

BINOMIAL THEOREM FOR NEGATIVE / FRACTIONAL EXPONENT (used in JEE Main rarely, Advanced more often):

For |x| < 1 and any real n:

(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + ...

This is an infinite series (no terminates).

Useful series:

(1 + x)⁻¹ = 1 − x + x² − x³ + ... (geometric series)
(1 − x)⁻¹ = 1 + x + x² + x³ + ...
(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + ...
(1 − x)⁻² = 1 + 2x + 3x² + 4x³ + ...
(1 + x)^(1/2) = 1 + x/2 − x²/8 + x³/16 − ... (for binomial approximation in physics)

Properties of binomial coefficients

Pascal's identity, sum of coefficients.