Official solution
\[\sum\limits_{n = 1}^{21} {{3 \over {(4n - 1)(4n + 3)}} = {3 \over 4}\sum\limits_{n = 1}^{21} {{1 \over {4n - 1}} - {1 \over {4n + 3}}} } \]
\[ = {3 \over 4}\left[ {\left( {{1 \over 3} - {1 \over 7}} \right) + \left( {{1 \over 7} - {1 \over {11}}} \right) + \,\,....\,\, + \,\,\left( {{1 \over {83}} - {1 \over {87}}} \right)} \right]\]
\[ = {3 \over 4}\left[ {{1 \over 3} - {1 \over {87}}} \right] = {3 \over 4}{{84} \over {3.87}} = {7 \over {29}}\]