Question

Câu 13 (0,5 điểm). Tìm x biết:
\[
\left| x + \frac{1}{1 \cdot 2} \right| + \left| x + \frac{1}{2 \cdot 3} \right| + \ldots + \left| x + \frac{1}{2024 \cdot 2025} \right| = 2025x
\]

Answer 1

Ta có: \(\left|x+\frac{1}{1\cdot2}\right|+\left|x+\frac{1}{2\cdot3}\right|+\ldots+\left|x+\frac{1}{2024\cdot2025}\right|=2025x\)

=>2025x>=0

=>x>=0

=>\(x+\frac{1}{1\cdot2}>0;x+\frac{1}{2\cdot3}>0;\ldots;x+\frac{1}{2024\cdot2025}>0\)

Phương trình sẽ trở thành

\(2025x=x+\frac{1}{1\cdot2}+x+\frac{1}{2\cdot3}+\cdots+x+\frac{1}{2024\cdot2025}\)

=>\(2025x=2024x+\left(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\cdots+\frac{1}{2024\cdot2025}\right)\)

=>\(x=\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\cdots+\frac{1}{2024\cdot2025}\)

=>\(x=1-\frac12+\frac12-\frac13+\cdots+\frac{1}{2024}-\frac{1}{2025}=1-\frac{1}{2025}=\frac{2024}{2025}\)