Formula Sheet

Binomial Theorem Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

8 formulas · 7 subtopics

(a+b)^n = \sum_{r=0}^{n} {}^nC_r a^{n-r} b^r

💡 n must be a non-negative integer. Total (n+1) terms.

T_{r+1} = {}^nC_r a^{n-r} b^r

💡 The (r+1)th term. r starts from 0. For finding a specific coefficient, set the power of x equal to the required value and solve for r.

⚠ Confusing T_r with T_{r+1}. The general term formula gives T_{r+1}, not T_r.

\text{If } n \text{ even: } T_{n/2+1} \quad | \quad \text{If } n \text{ odd: } T_{(n+1)/2} \text{ and } T_{(n+3)/2}

💡 Even n: one middle term. Odd n: two middle terms. The middle term often has the greatest binomial coefficient.

\sum_{r=0}^{n} {}^nC_r = 2^n

💡 Put x=1 in (1+x)^n. Sum of all coefficients of (1+x)^n is 2^n.

\sum_{r=0}^{n} (-1)^r {}^nC_r = 0

💡 Put x=-1 in (1+x)^n. Even-indexed and odd-indexed coefficients have equal sum.

\text{Sum of coefficients of } f(x) = f(1)

💡 To find sum of all coefficients in any expansion, substitute x=1. Works for any polynomial expression.

⚠ Confusing sum of binomial coefficients (always 2^n) with sum of all coefficients (put x=1 in the full expression).

T_{r+1} = {}^nC_r a^{(n-r)/p} b^{r/q} \text{ is integral when } p \mid (n-r) \text{ and } q \mid r

💡 For (a^{1/p} + b^{1/q})^n, the term is rational only when both exponents are integers.

a^n = (m \pm k)^n \text{, expand and isolate remainder from } k^n

💡 Write the base as (multiple of divisor ± small number). All terms except the last are divisible.