Question

Chứng minh rằng: 1/1.2.3+1/2.3.4+1/34.5+.....+1/18.19.20<1/4

Answer 1

Nhận thấy: 1�⋅(�+1)⋅(�+2)=22⋅�⋅(�+1)⋅(�+2)=2+�−�2�⋅(�+1)⋅(�+2)=12⋅[2+�−��⋅(�+1)⋅(�+2)]=12⋅[2+��⋅(�+1)⋅(�+2)−��⋅(�+1)⋅(�+2)]=12⋅[1�⋅(�+1)−1(�+1)⋅(�+2)]n⋅(n+1)⋅(n+2)1​=2⋅n⋅(n+1)⋅(n+2)2​=2n⋅(n+1)⋅(n+2)2+n−n​=21​⋅[n⋅(n+1)⋅(n+2)2+n−n​]=21​⋅[n⋅(n+1)⋅(n+2)2+n​−n⋅(n+1)⋅(n+2)n​]=21​⋅[n⋅(n+1)1​−(n+1)⋅(n+2)1​]

⇒�=11⋅2⋅3+12⋅3⋅4+...+118⋅19⋅20=12⋅[11⋅2−12⋅3+12⋅3−13⋅4+...+118⋅19−119⋅20]=12⋅[11⋅2−119⋅20]=14−1760<14⇒A=1⋅2⋅31​+2⋅3⋅41​+...+18⋅19⋅201​=21​⋅[1⋅21​−2⋅31​+2⋅31​−3⋅41​+...+18⋅191​−19⋅201​]=21​⋅[1⋅21​−19⋅201​]=41​−7601​<41​

Vậy �<14A<41​