Question

Tìm x biết:

\(\frac{1}{1.3}\)+\(\frac{1}{3.5}\)+ \(\frac{1}{5.7}\)+......+\(\frac{1}{x+\left(x+2\right)}\)=\(\frac{20}{21}\)

Answer 1

1/1.3 + 1/3.5 + ... + 1/x(x+2) = 20/21 

=>  2/1.3 + 2/3.5  + ... + 2/x(x+2) = 20/21

1 - 1/3 + 1/3 - 1/5 + ... + 1/x - 1/x+2 = 20/21

1 - 1/x+2 = 20/21

1/x+2 = 1/21

=> x + 2 = 21

=> x = 19

Answer 2

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Answer 3

\(\Rightarrow\frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+.......+\frac{1}{x}-\frac{1}{x+2}\right)=\frac{20}{21}\)

\(\Rightarrow1-\frac{1}{x+2}=\frac{20}{21}:\frac{1}{2}\)

\(\Rightarrow\frac{x+1}{x+2}=\frac{40}{21}\)

=>(x+1)21=(x+2)40

=>21x+21=40x+80

=>61x=101

=>x\(\in\varphi\)