Question

[Đỗ Ngọc Khánh Ly]

Tính H=\(2012-\left(1+\frac{1}{1+2}+\frac{1}{1+2+3}+...+\frac{1}{1+2+3+...+99}\right)\)

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giải cả lời giải nữa nhé

Answer 1

[Phùng Quang Thịnh]

H = 2012 - 1 - ( \(\frac{1}{1+2}+\frac{1}{1+2+3}+...+\frac{1}{1+2+...+99}\))
   = 2011 - ( \(\frac{1}{3}+\frac{1}{6}+...+\frac{1}{\left(99+1\right).\left[\left(99-1\right):1+1\right]:2}\)
   = 2011 - ( \(\frac{1}{3}+\frac{1}{6}+...+\frac{1}{4950}\))
   = 2011 - 2.( \(\frac{1}{6}+\frac{1}{12}+...+\frac{1}{9900}\))
   = 2011 - 2.(\(\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\)) 
   = 2011 - 2.( \(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\))
   = 2011 - 2.(\(\frac{1}{2}-\frac{1}{100}\)) = 2011 - 2.\(\frac{49}{100}\)= 2011 - \(\frac{49}{50}\)= \(\frac{100501}{50}\)

Answer 2

[ST]

\(H=2012-\left(1+\frac{1}{1+2}+\frac{1}{1+2+3}+...+\frac{1}{1+2+3+...+99}\right)\)

\(=2012-\left(1+\frac{1}{2\left(2+1\right):2}+\frac{1}{3\left(3+1\right):2}+...+\frac{1}{99\left(99+1\right):2}\right)\)

\(=2012-\left(\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+...+\frac{2}{99.100}\right)\)

\(=2012-2\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{2}{99.100}\right)\)

\(=2012-2\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\right)\)

\(=2012-2\left(1-\frac{1}{100}\right)\)

\(=2012-2\cdot\frac{99}{100}\)

\(=2012-\frac{99}{50}\)

\(=\frac{100501}{50}\)