Question

[Korea Yogurt Yogun]

1/15+1/21+1/28 +... 2/Xx(X+1)=1/3

Answer 1

[Dang Tung]

\(\dfrac{1}{15}+\dfrac{1}{21}+\dfrac{1}{28}+...+\dfrac{2}{x\times\left(x+1\right)}=\dfrac{1}{3}\\ \dfrac{2}{30}+\dfrac{2}{42}+\dfrac{2}{56}+...+\dfrac{2}{x\times\left(x+1\right)}=\dfrac{1}{3}\\ 2\times\left(\dfrac{1}{5\times6}+\dfrac{1}{6\times7}+\dfrac{1}{7\times8}+...+\dfrac{1}{x\times\left(x+1\right)}\right)=\dfrac{1}{3}\\ \dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{8}+...+\dfrac{1}{x}-\dfrac{1}{x+1}=\dfrac{1}{3}:2\\ \dfrac{1}{5}-\dfrac{1}{x+1}=\dfrac{1}{6}\\ \dfrac{1}{x+1}=\dfrac{1}{5}-\dfrac{1}{6}=\dfrac{1}{30}\\ x+1=30\\ x=29\)

Answer 2

[Akai Haruma]

Lời giải:

** Sửa đề:

$\frac{1}{15}+\frac{1}{21}+\frac{1}{28}+...+\frac{2}{x(x+1)}=\frac{1}{3}$

$\frac{2}{30}+\frac{2}{42}+\frac{2}{56}+...+\frac{2}{x(x+1)}=\frac{1}{3}$

$\frac{2}{5.6}+\frac{2}{6.7}+\frac{2}{7.8}+...+\frac{2}{x(x+1)}=\frac{1}{3}$

$2(\frac{6-5}{5.6}+\frac{7-6}{6.7}+\frac{8-7}{7.8}+...+\frac{x+1-x}{x(x+1)})=\frac{1}{3}$

$2(\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+...+\frac{1}{x}-\frac{1}{x+1})=\frac{1}{3}$

$2(\frac{1}{5}-\frac{1}{x+1})=\frac{1}{3}$

$\frac{1}{5}-\frac{1}{x+1}=\frac{1}{6}$

$\frac{1}{x+1}=\frac{1}{5}-\frac{1}{6}=\frac{1}{30}$

$\Rightarrow x+1=30$

$\Rightarrow x=29$.