Câu hỏi của Trần Thảo Anh

Question

Tính: 2/9 + 2/45 + 2/105 + 2/189 +...+ 2/1725

Nguyễn Lê Phước Thịnh · Accepted Answer

$\frac29+\frac{2}{45}+\frac{2}{105}+\frac{2}{189}+\cdots+\frac{2}{1725}$

$=\frac23\left(\frac13+\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\cdots+\frac{1}{575}\right)$

$=\frac23\times\frac12\times\left(\frac{2}{1\times3}+\frac{2}{3\times5}+\ldots+\frac{2}{23\times25}\right)$

$=\frac13\times\left(1-\frac13+\frac13-\frac15+\cdots+\frac{1}{23}-\frac{1}{25}\right)=\frac13\times\left(1-\frac{1}{25}\right)$

$=\frac13\times\frac{24}{25}=\frac{8}{25}$

Câu trả lời:

$\frac29+\frac{2}{45}+\frac{2}{105}+\frac{2}{189}+\cdots+\frac{2}{1725}$

$=\frac23\left(\frac13+\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\cdots+\frac{1}{575}\right)$

$=\frac23\times\frac12\times\left(\frac{2}{1\times3}+\frac{2}{3\times5}+\ldots+\frac{2}{23\times25}\right)$

$=\frac13\times\left(1-\frac13+\frac13-\frac15+\cdots+\frac{1}{23}-\frac{1}{25}\right)=\frac13\times\left(1-\frac{1}{25}\right)$

$=\frac13\times\frac{24}{25}=\frac{8}{25}$