Ta có :
a) \(\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\frac{1}{99}+\frac{1}{143}+\frac{1}{195}\)
\(=\)\(\frac{1}{2}\left(\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+\frac{2}{9.11}+\frac{2}{11.13}+\frac{2}{13.15}\right)\)
\(=\)\(\frac{1}{2}\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}+\frac{1}{11}-\frac{1}{13}+\frac{1}{13}-\frac{1}{15}\right)\)
\(=\)\(\frac{1}{2}\left(\frac{1}{3}-\frac{1}{15}\right)\)
\(=\)\(\frac{1}{2}.\frac{4}{15}\)
\(=\)\(\frac{2}{15}\)
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 :
\(\frac{1+13+25+37+...+229}{1000}=\frac{2300}{1000}=\frac{23}{10}\)