Binomial Theorem — Formula Sheet

(x + y)^n = \sum_{r=0}^{n} {}^nC_r x^{n-r} y^r

T_{r+1} = {}^nC_r x^{n-r} y^r

Middle Term = T_{\frac{n}{2} + 1} \text{ (n even)}

Middle Terms = T_{\frac{n+1}{2}}, T_{\frac{n+3}{2}} \text{ (n odd)}

(1 + x)^r = 1 + rx + \frac{r(r-1)}{2!}x^2 + \frac{r(r-1)(r-2)}{3!}x^3 + \dots

(x + y)^r = x^r \left(1 + \frac{y}{x}\right)^r, \left|\frac{y}{x}\right| < 1

(1 + x)^n \approx 1 + nx \text{ (small x)}