- Với \(n=1\Rightarrow1.2.3=\frac{1.2.3.4}{4}\) (đúng)
- Giả sử biểu thức đúng với \(n=k\) hay:
\(1.2.3+...+k\left(k+1\right)\left(k+2\right)=\frac{k\left(k+1\right)\left(k+2\right)\left(k+3\right)}{4}\)
Ta cần chứng minh nó đúng với \(n=k+1\) hay:
\(1.2.3+...+k\left(k+1\right)\left(k+2\right)+\left(k+1\right)\left(k+2\right)\left(k+3\right)=\frac{\left(k+1\right)\left(k+2\right)\left(k+3\right)\left(k+4\right)}{4}\)
Thật vậy, ta có:
\(1.2.3+...+k\left(k+1\right)\left(k+2\right)+\left(k+1\right)\left(k+2\right)\left(k+3\right)\)
\(=\frac{k\left(k+1\right)\left(k+2\right)\left(k+3\right)}{4}+\left(k+1\right)\left(k+2\right)\left(k+3\right)\)
\(=\left(k+1\right)\left(k+2\right)\left(k+3\right)\left[\frac{k}{4}+1\right]\)
\(=\left(k+1\right)\left(k+2\right)\left(k+3\right).\frac{\left(k+4\right)}{4}\)
\(=\frac{\left(k+1\right)\left(k+2\right)\left(k+3\right)\left(k+4\right)}{4}\) (đpcm)
