Question

[Lê Tiến Dũng]

tính nhanh 

a}1/15 + 1/35 + 1/63 + 1/99 + 1/143 + 1/195                                     b}1/3 + 1/6 + 1/12 + 1/24 + 1/48 +1/96

c}1/1000 + 13/1000 + 25/1000 + 37/1000 +....+229/1000                 d}2012 x 2014 -1/2011 + 2012 + 2013 x 7/5

Answer 1

[Phùng Minh Quân]

Ta có :

a) \(\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\frac{1}{99}+\frac{1}{143}+\frac{1}{195}\)

\(=\)\(\frac{1}{2}\left(\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+\frac{2}{9.11}+\frac{2}{11.13}+\frac{2}{13.15}\right)\)

\(=\)\(\frac{1}{2}\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}+\frac{1}{11}-\frac{1}{13}+\frac{1}{13}-\frac{1}{15}\right)\)

\(=\)\(\frac{1}{2}\left(\frac{1}{3}-\frac{1}{15}\right)\)

\(=\)\(\frac{1}{2}.\frac{4}{15}\)

\(=\)\(\frac{2}{15}\)

Answer 2

[Phùng Minh Quân]

Ta có : 

\(c)\)\(\frac{1}{1000}+\frac{13}{1000}+\frac{25}{1000}+\frac{37}{1000}+...+\frac{229}{1000}\)

\(=\)\(\frac{1+13+25+37+...+229}{1000}\)

Xét tổng \(1+13+25+37+...+229\):

Số số hạng : \(\left(229-1\right):12+1=20\) ( số hạng )

Tổng : \(\frac{\left(229+1\right).20}{2}=2300\)

Do đó :

\(\frac{1+13+25+37+...+229}{1000}=\frac{2300}{1000}=\frac{23}{10}\)