JEE Advance - Mathematics (1993 - No. 11)
Using mathematical induction, prove that
$${\tan ^{ - 1}}\left( {1/3} \right) + {\tan ^{ - 1}}\left( {1/7} \right) + ........{\tan ^{ - 1}}\left\{ {1/\left( {{n^2} + n + 1} \right)} \right\} = {\tan ^{ - 1}}\left\{ {n/\left( {n + 2} \right)} \right\}$$
$${\tan ^{ - 1}}\left( {1/3} \right) + {\tan ^{ - 1}}\left( {1/7} \right) + ........{\tan ^{ - 1}}\left\{ {1/\left( {{n^2} + n + 1} \right)} \right\} = {\tan ^{ - 1}}\left\{ {n/\left( {n + 2} \right)} \right\}$$
The statement is false for all n.
The statement is true for n = 1, but the inductive step fails.
The base case holds and the inductive step can be proven, therefore the statement is true for all n.
The statement is true for n = 1, but the statement is false for n > 1
Mathematical induction is not applicable to this type of problem.
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