1) tính nhanh S = 1 + 1/ 1 +2 + 1/1+2+3 + 1/1+2+3+4 +…+1/1+2+3+4 + …+ 8

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1. Đáp án:

    $S=\frac{16}{9}$

   Giải thích các bước giải:

    $S=1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}…+\frac{1}{1+2+3+…+8}\\
   =1+\frac{1}{2.3:2}+\frac{1}{3.4:2}+\frac{1}{4.5:2}+…+\frac{1}{8.9:2}\\
   =1+2\left (\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+…+\frac{1}{8.9}  \right )\\
   =1+2\left ( \frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+…+\frac{1}{8}-\frac{1}{9}\right )\\
   =1+2\left ( \frac{1}{2}-\frac{1}{9} \right )\\
   =1+2\left ( \frac{9}{18}-\frac{2}{18} \right )\\
   =1+2.\frac{7}{18}\\
   =1+\frac{7}{9}\\
   =\frac{9+7}{9}=\frac{16}{9}$

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