Question

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Tính nhanh tổng dãy phân số sau :

1/3 + 1/6 + 1/10 + .....+ 2/x(x + 1) = 499/999

Answer 1

[Đức Phạm]

\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+....+\frac{2}{x\left(x+1\right)}=\frac{499}{999}\)

\(\Leftrightarrow\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+....+\frac{2}{x\left(x+1\right)}=\frac{499}{999}\)

\(\Leftrightarrow\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+....+\frac{2}{x\left(x+1\right)}=\frac{499}{999}\)

\(\Leftrightarrow2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+....+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{499}{999}\)

\(\Leftrightarrow\frac{1}{2}-\frac{1}{x+1}=\frac{499}{999}\div2\)

\(\Leftrightarrow\frac{1}{x+1}=\frac{1}{2}-\frac{499}{1998}\)

\(\Leftrightarrow\frac{1}{x+1}=\frac{250}{999}\)

\(\Leftrightarrow\left(x+1\right).250=999\Rightarrow x+1=\frac{999}{250}\Rightarrow x=\frac{999}{250}-1=\frac{749}{250}\)

Như kiểu đề sai hay sao ý