Binomial Theorem formulas

Formulas

Binomial theorem

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

General term

$\displaystyle T_{r + 1} = {}^nC_r\,a^{n - r}b^r$

Number of terms

$\displaystyle n + 1$

Sum of coefficients

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

Sum of odd and even coefficients

$\displaystyle {}^nC_0 + {}^nC_2 + \cdots = {}^nC_1 + {}^nC_3 + \cdots = 2^{n - 1}$

Middle term

• If n is even, the one middle term is term number (n/2 + 1).
• If n is odd, there are two middle terms: numbers (n + 1)/2 and (n + 3)/2.