• # question_answer The value of $\frac{3}{{{1}^{1}}{{.2}^{2}}}+\frac{5}{{{2}^{2}}{{.3}^{2}}}+\frac{7}{{{3}^{2}}{{.4}^{2}}}+\frac{9}{{{4}^{2}}{{.5}^{2}}}+\frac{11}{{{5}^{2}}{{.6}^{2}}}+$$\frac{13}{{{6}^{2}}{{.7}^{2}}}+\frac{15}{{{7}^{2}}{{.8}^{2}}}+\frac{17}{{{8}^{2}}{{.9}^{2}}}+\frac{19}{{{9}^{2}}{{.10}^{2}}}$ is A)  $\frac{1}{100}$                                   B)  $\frac{99}{100}$ C)  $\frac{101}{100}$                               D)  1

Expression $=\frac{3}{{{1}^{2}}{{.2}^{2}}}+\frac{5}{{{2}^{2}}{{.3}^{2}}}+\frac{7}{{{3}^{2}}{{.4}^{2}}}+....+\frac{17}{{{8}^{2}}{{.9}^{2}}}+\frac{19}{{{9}^{2}}{{.10}^{2}}}$ $=\left( \frac{1}{{{1}^{2}}}-\frac{1}{{{2}^{2}}} \right)+\left( \frac{1}{{{2}^{2}}}-\frac{1}{{{3}^{2}}} \right)+\left( \frac{1}{{{3}^{2}}}-\frac{1}{{{4}^{2}}} \right)$ $...+\left( \frac{1}{{{8}^{2}}}-\frac{1}{{{9}^{2}}} \right)+\left( \frac{1}{{{9}^{2}}}-\frac{1}{{{10}^{2}}} \right)$ $=\frac{1}{{{1}^{2}}}-\frac{1}{{{2}^{2}}}+\frac{1}{{{2}^{2}}}-\frac{1}{{{3}^{2}}}+\frac{1}{{{3}^{2}}}-\frac{1}{{{4}^{2}}}+......$ $\frac{1}{{{8}^{2}}}-\frac{1}{{{9}^{2}}}+\frac{1}{{{9}^{2}}}-\frac{1}{{{10}^{2}}}$ $=1-\frac{1}{{{10}^{2}}}$ $=1-\frac{1}{100}=\frac{100-1}{100}=\frac{99}{100}$