| Version 8 |
Version 7 |
| The binomial theorem is a formula for the complete expansion of $(a+b)^n, n \geq 1 ,$ into a sum of powers of $a$ and $b$. More precisely, |
The binomial theorem is a formula for the complete expansion of $(a+b)^n, n \geq 1 ,$ into a sum of powers of $a$ and $b$. More precisely, |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| (a+b)^n & = \sum_{k=0}^n \binom{n}{k} a^{n-k}b^k \\ |
(a+b)^n & = \sum_{k=0}^n \binom{n}{k} a^{n-k}b^k \\ |
|
& = a^n + \binom{n}{1} a^{n-1}b + \binom{n}{2} a^{n-2}b^2 + \cdots + b^n .
|
& = a^n + \binom{n}{1} a^{n-1}b + \binom{n}{2} a^{n-2}b^2 + \cdots \ b^n .
|
| \end{eqnarray*} |
\end{eqnarray*} |
| For example, if $n=3$, we have: |
For example, if $n=3$, we have: |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| (a+b)^3 &= a^3 + 3 a^2 b + 3 a b^2 + b^3 \\ |
(a+b)^3 &= a^3 + 3 a^2 b + 3 a b^2 + b^3 \\ |
| (a+b)^4 &= a^4 + 4 a^3 b + 6 a^2 b^2 + 4 a b^3 + b^4 . |
(a+b)^4 &= a^4 + 4 a^3 b + 6 a^2 b^2 + 4 a b^3 + b^4 . |
| \end{eqnarray*} |
\end{eqnarray*} |
