A = \(\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{217.218}\)
A = \(\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{217}-\dfrac{1}{218}\)
A = 1 - \(\dfrac{1}{218}\)
B = \(\dfrac{1}{110}\) + \(\dfrac{1}{111}\) + \(\dfrac{1}{112}\) + ... + \(\dfrac{1}{218}\)
Xét dãy số 110; 111; 112; ...; 218, dãy số này có số số hạng là:
(218 - 110) : 1 + 1 = 109 (số)
Mặt khác \(\dfrac{1}{110}\) > \(\dfrac{1}{111}>\dfrac{1}{112}>...>\dfrac{1}{218}\)
⇒ B = \(\dfrac{1}{110}\) + \(\dfrac{1}{111}\) + \(\dfrac{1}{112}+...+\dfrac{1}{218}\) < \(\dfrac{1}{110}\) + \(\dfrac{1}{110}\)+ ... +\(\dfrac{1}{110}\)
B < \(\dfrac{1}{110}\) x 109
B < 1 - \(\dfrac{1}{110}\)
\(\dfrac{1}{128}\) < \(\dfrac{1}{110}\) ⇒ A = 1 - \(\dfrac{1}{128}\) > 1 - \(\dfrac{1}{110}\) > B
A > B
