Chứng tỏ rằng A<5/4 A=3/1.2.3+5/2.3.4+7/3.4.5+....+201/100.101.102

Đáp án:

$$A = {{12675} \over {10302}} < {5 \over 4}$$

Giải thích các bước giải:

$$\eqalign{ & A = {3 \over {1.2.3}} + {5 \over {2.3.4}} + {7 \over {3.4.5}} + ... + {{201} \over {100.101.102}} \cr & A = \left( {{1 \over {1.2.3}} + {1 \over {2.3.4}} + {1 \over {3.4.5}} + ... + {1 \over {100.101.102}}} \right) + \left( {{2 \over {1.2.3}} + {4 \over {2.3.4}} + {6 \over {3.4.5}} + ... + {{200} \over {100.101.102}}} \right) \cr & A = \left( {{1 \over {1.2.3}} + {1 \over {2.3.4}} + {1 \over {3.4.5}} + ... + {1 \over {100.101.102}}} \right) + 2\left( {{1 \over {1.2.3}} + {2 \over {2.3.4}} + {3 \over {3.4.5}} + ...{{100} \over {100.101.102}}} \right) \cr & A = B + 2C \cr & B = {1 \over {1.2.3}} + {1 \over {2.3.4}} + {1 \over {3.4.5}} + ... + {1 \over {100.101.102}} \cr & 2B = {2 \over {1.2.3}} + {2 \over {2.3.4}} + {2 \over {3.4.5}} + ... + {2 \over {100.101.102}} \cr & 2B = \left( {{1 \over {1.2}} - {1 \over {2.3}} + {1 \over {2.3}} - {1 \over {3.4}} + {2 \over {3.4}} - {2 \over {4.5}} + ... + {1 \over {100.101}} - {1 \over {101.102}}} \right) \cr & 2B = {1 \over {1.2}} - {1 \over {101.102}} = {{2575} \over {5151}} \cr & B = {{2575} \over {5151}}:2 = {{2575} \over {10302}} \cr & C = {1 \over {1.2.3}} + {2 \over {2.3.4}} + {3 \over {3.4.5}} + ...{{100} \over {100.101.102}} \cr & C = {1 \over {2.3}} + {1 \over {3.4}} + {1 \over {4.5}} + ... + {1 \over {101.102}} \cr & C = {1 \over 2} - {1 \over 3} + {1 \over 3} - {1 \over 4} + ... + {1 \over {101}} - {1 \over {102}} \cr & C = {1 \over 2} - {1 \over {102}} = {{25} \over {51}} \cr & Vay\,\,A = B + 2C = {{2575} \over {10302}} + {{50} \over {51}} = {{2575 + 202.50} \over {5151}} = {{12675} \over {10302}} \cr & {{12675} \over {10302}} = {{12675.2} \over {10302.2}} = {{25350} \over {20604}} \cr & {5 \over 4} = {{5.5151} \over {4.5151}} = {{25755} \over {20604}} \cr & \Rightarrow {{12675} \over {10302}} < {5 \over 4} \Rightarrow A < {5 \over 4}\,\,\left( {dpcm} \right) \cr} $$