-4/1.2+ -4/2.3+-4/3.4+...+ -4/97.98+-4/98.99 Tham khảo câu trả lời từ Trợ lý ảo Lazi To simplify the expression given, we first notice that we have two identical series: \[ -4 \left( \frac{1}{1.2} + \frac{1}{2.3} + \frac{1}{3.4} + \ldots + \frac{1}{97.98} + \frac{1}{98.99} \right) \] We can denote the series inside the parentheses as \( S \): \[ S = \frac{1}{1.2} + \frac{1}{2.3} + \frac{1}{3.4} + \ldots + \frac{1}{97.98} + \frac{1}{98.99} \] The final expression then becomes: \[ -4S - 4S = -8S \] Now, we must calculate \( S \). ### Analyzing Terms of S The general term of \( S \) appears to be of the form: \[ \frac{1}{n(n+1)} \quad \text{where } n \text{ runs from 1 to 98.} \] ### Calculating S We can express \( S \) as follows: \[ S = \sum_{n=1}^{98} \frac{1}{n(n+1)} \] Using the technique of partial fractions, we have: \[ \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \] Thus, we can rewrite \( S \): \[ S = \sum_{n=1}^{98} \left( \frac{1}{n} - \frac{1}{n+1} \right) \] ### Telescoping Series This forms a telescoping series: \[ S = \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \cdots + \left( \frac{1}{98} - \frac{1}{99} \right) \] When we sum this, all intermediate terms will cancel out, leaving us with: \[ S = 1 - \frac{1}{99} \] Calculating this gives: \[ S = \frac{99}{99} - \frac{1}{99} = \frac{98}{99} \] ### Final Calculation Now substituting \( S \) back into our expression for the total: \[ -8S = -8 \left( \frac{98}{99} \right) = -\frac{784}{99} \] Thus, the final result of the entire expression is: \[ \boxed{-\frac{784}{99}} \]