\(S=\frac{3}{1\cdot4}+\frac{3}{4\cdot7}+\frac{3}{7\cdot11}+\frac{3}{11\cdot14}+\frac{3}{14\cdot17}\)
\(S=\frac{4-1}{1\cdot4}+\frac{7-4}{4\cdot7}+\frac{11-7}{7\cdot11}+\frac{14-11}{11\cdot14}+\frac{17-14}{14\cdot17}\)
\(S=\frac{4}{1\cdot4}-\frac{1}{1\cdot4}+\frac{7}{7\cdot4}-\frac{4}{4\cdot7}+\frac{11}{7\cdot11}-\frac{7}{7\cdot11}+\frac{14}{11\cdot14}-\frac{11}{11\cdot14}+\frac{17}{14\cdot17}-\frac{14}{14\cdot17}\)
\(S=\frac{1}{1}-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{14}+\frac{1}{14}-\frac{1}{17}\)
\(S=\frac{1}{1}-\frac{1}{17}=\frac{17}{17}-\frac{1}{17}=\frac{16}{17}\)
\(S=\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.11}+\frac{3}{11.14}+\frac{3}{14.17}\)
\(S=1.\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+...+\frac{1}{14}-\frac{1}{17}\right)\)
\(S=1.\left(1-\frac{1}{17}\right)\)
\(S=1.\frac{16}{17}\)
\(S=\frac{16}{17}\)
\(S=\frac{3}{1.4}+\frac{3}{4.7}+\frac{3}{7.11}+\frac{3}{11.14}+\frac{3}{14.17}\)
\(S=\frac{4-1}{1.4}+\frac{7-4}{4.7}+\frac{11-7}{7.11}+\frac{14-11}{11.14}+\frac{17-14}{14.17}\)
\(S=\frac{4}{1.4}-\frac{1}{1.4}+\frac{7}{4.7}-\frac{4}{7.4}+\frac{11}{7.11}-\frac{7}{7.11}+\frac{14}{11.14}-\frac{11}{11.14}+\frac{17}{17.14}-\frac{14}{17.14}\)
\(S=1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{14}+\frac{1}{14}-\frac{1}{17}\)
\(S=1-\frac{1}{17}=\frac{16}{17}\)
