\(\frac{1}{6}+\frac{1}{66}+\frac{1}{176}+...+\frac{1}{248496}\)
\(=\frac{1}{1.6}+\frac{1}{6.11}+\frac{1}{11.16}+....+\frac{1}{496.501}\)
\(=\frac{1}{5}.\left(\frac{5}{1.6}+\frac{5}{6.11}+...+\frac{5}{496.501}\right)\)
\(=\frac{1}{5}.\left(1-\frac{1}{6}+\frac{1}{6}-\frac{1}{11}+...+\frac{1}{496}-\frac{1}{501}\right)\)
\(=\frac{1}{5}.\left(1-\frac{1}{501}\right)\)
\(=\frac{1}{5}.\frac{500}{501}\)
\(=\frac{100}{501}\)
1/6 + 1/66 + 1/176 + ... + 1/248496
= 1/1*6 + 1/6*11 + 1/11*16 + ... + 1/496*501
= 1/5(5/1*6 + 5/6*11 + 5/11*16 + ... + 5/496*501)
= 1/5(1 - 1/6 + 1/6 - 1/11 + 1/11 - 1/16 + ... + 1/496 - 1/501)
= 1/5(1 - 1/501)
\(\frac{1}{6}+\frac{1}{66}+\frac{1}{176}+...+\frac{1}{248496}\)
\(=\frac{1}{1\times6}+\frac{1}{6\times11}+...+\frac{1}{496\times501}\)
\(=\frac{1}{5}\left(\frac{5}{1\times6}+\frac{5}{6\times11}+...+\frac{5}{496\times501}\right)\)
\(=\frac{1}{5}\left(1-\frac{1}{6}+\frac{1}{6}-\frac{1}{11}+\frac{1}{11}-\frac{1}{16}+...+\frac{1}{496}-\frac{1}{501}\right)\)
\(=\frac{1}{5}\left(1-\frac{1}{501}\right)=\frac{1}{5}\times\frac{500}{501}=\frac{100}{501}\)
