Question

Chứng tỏ:

1/1.2.3+1/2.​3.4+1/3.4.5+...+1/10.11.12 <1/4

Chỉ giùm cách trình bày luon nha!thanks!

Answer 1

\(\frac{1}{\left(k-1\right)k\left(k+1\right)}=\frac{k+1-\left(k-1\right)}{\left(k-1\right)k\left(k+1\right)}\frac{1}{2}\)\(=\frac{1}{2}\left(\frac{k+1}{\left(k-1\right)k\left(k+1\right)}-\frac{k-1}{\left(k-1\right)k\left(k+1\right)}\right)\)\(=\frac{1}{2}\left(\frac{1}{\left(k-1\right)k}-\frac{1}{k\left(k+1\right)}\right)\)

\(\Rightarrow\)VT\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{10.11}-\frac{1}{11.12}\right)\)\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{11.12}\right)\)