The value of $$\frac{3}{{{1^2} \times {2^2}}} + $$   $$\frac{5}{{{2^2} \times {3^2}}} + $$   $$\frac{7}{{{3^2} \times {4^2}}} + $$   $$\frac{9}{{{4^2} \times {5^2}}} + $$   $$\frac{{11}}{{{5^2} \times {6^2}}} + $$   $$\frac{{13}}{{{6^2} \times {7^2}}} + $$   $$\frac{{15}}{{{7^2} \times {8^2}}} + $$   $$\frac{{17}}{{{8^2} \times {9^2}}} + $$   $$\frac{{19}}{{{9^2} \times {{10}^2}}}$$   is = ?

Solution (By Examveda Team)

$$\eqalign{ & {\text{Given expression,}} \cr & \left( {\frac{1}{{{1^2}}} - \frac{1}{{{2^2}}}} \right) + \left( {\frac{1}{{{2^2}}} - \frac{1}{{{3^2}}}} \right) + \left( {\frac{1}{{{3^2}}} - \frac{1}{{{4^2}}}} \right) + \left( {\frac{1}{{{4^2}}} - \frac{1}{{{5^2}}}} \right) . . . . + \left( {\frac{1}{{{9^2}}} - \frac{1}{{{{10}^2}}}} \right) \cr & = \left( {\frac{1}{{{1^2}}} - \frac{1}{{{2^2}}}} \right) \cr & = \left( {1 - \frac{1}{{100}}} \right) \cr & = \frac{{99}}{{100}} \cr} $$