First slide

Properties of ITF

Question

If $a_{1}, a_{2}, a_{3}, ....... a_{n}$ is an AP with common difference d, then $\tan [\tan^{- 1} (\frac{d}{1 + a_{1} a_{2}}) + \tan^{- 1} (\frac{d}{1 + a_{2} a_{3}}) + .... + \tan^{- 1} (\frac{d}{1 + a_{n - 1} a_{n}})]$ is equal to

Moderate

A

$\frac{(n - 1) d}{a_{1} + a_{n}}$
B

$\frac{(n - 1) d}{1 + a_{1} a_{n}}$
C

$\frac{n d}{1 + a_{1} a_{n}}$
D

$\frac{a_{n} - a_{1}}{a_{n} + a_{1}}$

Solution

$W e h a v e \tan^{- 1} (\frac{a_{2} - a_{1}}{1 + a_{1} a_{2}}) + \tan^{- 1} (\frac{a_{3} - a_{2}}{1 + a_{2} a_{3}}) + ...... + \tan^{- 1} (\frac{a_{n} - a_{n - 1}}{1 + a_{n - 1} a_{n}}) (∵ d = a_{n} - a_{n - 1} \forall n)$
$= (\tan^{- 1} a_{2} - \tan^{- 1} a_{1}) + (\tan^{- 1} a_{3} - \tan^{- 1} a_{2}) + ...... + (\tan^{- 1} a_{n} - \tan^{- 1} a_{n - 1})$
$= \tan^{- 1} a_{n} - \tan^{- 1} a_{1} = \tan^{- 1} (\frac{a_{n} - a_{1}}{1 + a_{n} a_{1}})$
$= \tan^{- 1} (\frac{(n - 1) d}{1 + a_{1} a_{n}})$
$∴ \tan [\tan^{- 1} (\frac{d}{1 + a_{1} a_{2}}) + \tan^{- 1} (\frac{d}{1 + a_{2} a_{3}}) + .... + \tan^{- 1} (\frac{d}{1 + a_{n - 1} a_{n}})] = \frac{(n - 1) d}{1 + a_{1} a_{n}}$

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