Refresher - Binomial Theorem

Refresher

A binomial is a sum ${\displaystyle x+y}$ where x and y represent numbers. If n is a positive integer, then a general formula for expanding ${\displaystyle {(x+y)}^n}$ is given by the binomial theorem.

1. The following formula is true for each of the first n terms of the expansion:
   • ${\displaystyle \frac{\left(\text{coefficient of term}\right)\left(\text{exponent of a}\right)}{\left(\text{number of term}\right)}=\text{coefficient of next term}}$
      
2. The following formulas are valid for ${\displaystyle {(a+b)}^n}$ for an arbitrary positive integer n.
   • Coefficient of the k+1)st term in the expansion of  ${\displaystyle {(a+b)}^n}$
     ${\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)=C(n,k)=\frac{n!}{k!(n-k)!},k=0,1,2,3,\dots .n}$
   • The Binomial Theorem
     ${\displaystyle {(a+b)}^n=a^n+\left(\begin{array}{c}n\\ 1\end{array}\right)a^{n-1}{b}^{\mathrm{}}+...+\left(\begin{array}{c}n\\ k\end{array}\right)a^{n-k}{b}^k+...+\left(\begin{array}{c}n\\ n-1\end{array}\right)a{b}^{n-1}+b^n}$

Using the summation notation, we may write the binomial theorem

${\displaystyle {(a+b)}^n=\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)a^{n-k}{b}^k}$

Note: There are ${\displaystyle (n+1)\text{(not n terms})}$ terms in the expansion of  ${\displaystyle {(a+b)}^n}$, and so
${\displaystyle \left(\begin{array}{c}n\\ k\end{array}\right)a^{n-k}{b}^k}$ is a formula for the ${\displaystyle (k+1)}$st term of the expansion.