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# If ${a_1},{a_2},{a_3}, \ldots ,{a_n}$ is an arithmetic progression with common difference<br>$d$, then evaluate the following Exemplarpression.<br>$\tan \left[ {{{\tan }^{ – 1}}\left( {\frac{d}{{1 + {a_1}{a_2}}}} \right) + {{\tan }^{ – 1}}\left( {\frac{d}{{1 + {a_2}{a_3}}}} \right) + {{\tan }^{ – 1}}\left( {\frac{d}{{1 + {a_3}{a_4}}}} \right)} \right.$ $\left. { + \ldots + {{\tan }^{ – 1}}\left( {\frac{d}{{1 + {a_{n – 1}}{a_n}}}} \right)} \right]$

We have, ${a_1} = a,{a_2} = a + d,{a_3} = a + 2d$
and $d = {a_2} – {a_1} = {a_3} – {a_2} = {a_4} – {a_3} = \ldots = {a_n} – {a_{n – 1}}$

# $= \tan \left[ {{{\tan }^{ – 1}}\frac{{{a_n} – {a_1}}}{{1 + {a_n} \cdot {a_1}}}} \right]$ $= \frac{{{a_n} – {a_1}}}{{1 + {a_n} \cdot {a_1}}}$

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