Tính nhanh : (1-1/2^2) x (1-1/3^2) x (1-1/4^2) x … x (1-1/100^2)

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1. Đáp án:

   $\frac{101}{200}$ 

   Giải thích các bước giải:

   $\frac{3}{4}$ . $\frac{8}{9}$ . $\frac{15}{16}$ ….. $\frac{9999}{10000}$

   = $\frac{1.3}{2.2}$ . $\frac{2.4}{3.3}$ . $\frac{3.5}{4.4}$ ….. $\frac{99.101}{100.100}$

   = $\frac{1.2.3…..99}{2.3.4…..100}$ . $\frac{3.4.5…..101}{2.3.4…..100}$

   = $\frac{1}{100}$ . $\frac{101}{2}$

   = $\frac{101}{200}$ 

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2. \(\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)…\left(1-\dfrac{1}{99^2}\right)\left(1-\dfrac{1}{100^2}\right)\)

   \(=\left(1-\dfrac{1}{2}\right)\left(1+\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\left(1+\dfrac{1}{3}\right)…\left(1-\dfrac{1}{99}\right)\left(1+\dfrac{1}{99}\right)\left(1-\dfrac{1}{100}\right)\left(1+\dfrac{1}{100}\right)\)

   \(=\dfrac{1}{2}.\dfrac{3}{2}.\dfrac{2}{3}.\dfrac{4}{3}…\dfrac{98}{99}.\dfrac{100}{99}.\dfrac{99}{100}.\dfrac{101}{100}\)

   \(=\dfrac{1.2…98.99}{2.3…99.100}.\dfrac{3.4…100.101}{2.3…99.100}\)

   \(=\dfrac{1}{100}.\dfrac{101}{2}=\dfrac{101}{200}\)

    đáp án đó bạn :))

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