If $a_{1}, a_{2}, a_{3}, \dots, a_{n}$ is an arithmetic progression with common difference $d$ , then evaluate the following expression
$tan ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} {tan}^{- 1} (\frac{d}{1 + a_{1} a_{2}}) + {tan}^{- 1} (\frac{d}{1 + a_{2} a_{3}}) + {tan}^{- 1} (\frac{d}{1 + a_{3} a_{4}}) \dots + {tan}^{- 1} (\frac{d}{1 + a_{n - 1} a_{n}}) + \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$

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Solution

Let
$A = tan ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} {tan}^{- 1} (\frac{d}{1 + a_{1} a_{2}}) + {tan}^{- 1} (\frac{d}{1 + a_{2} a_{3}}) + {tan}^{- 1} (\frac{d}{1 + a_{3} a_{4}}) \dots + {tan}^{- 1} (\frac{d}{1 + a_{n - 1} a_{n}}) \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$

$a_{1}, a_{2}, a_{3}, \dots, a_{n}$ is an $A P$
$\Rightarrow d = a_{2} - a_{1} = a_{3} - a_{2} = \dots = a_{n} - a_{n} - 1$

$\Rightarrow A = tan ⎡ ⎢ ⎢ ⎢ ⎣_{+ \dots + {tan}^{- 1} ⎛ ⎝ \frac{a_{n} - a_{n} - 1}{1 + a_{n} - 1 a_{n}} ⎞ ⎠}^{{tan}^{- 1} \frac{a_{2} - a_{1}}{1 + a_{1} a_{2}} + {tan}^{- 1} \frac{a_{3} - a_{2}}{1 + a_{3} a_{2}}} ⎤ ⎥ ⎥ ⎥ ⎦$

$\Rightarrow A = tan ⎡ ⎢ ⎢ ⎣ \begin{matrix} ({tan}^{- 1} a_{2} - {tan}^{- 1} a_{1}) + ({tan}^{- 1} a_{3} - {tan}^{- 1} a_{2}) + \dots + ({tan}^{- 1} a_{n} - {tan}^{- 1} a_{n - 1}) \end{matrix} ⎤ ⎥ ⎥ ⎦$
$[∵ {tan}^{- 1} x - {tan}^{- 1} y = {tan}^{- 1} \frac{x - y}{1} + x y, x y > - 1]$
$\Rightarrow A = tan ({tan}^{- 1} a_{n} - t a n^{- 1} a_{1})$
$\Rightarrow A = tan ({tan}^{- 1} \frac{a_{n} - a_{1}}{1 + a_{n} a_{1}})$
$\Rightarrow A = \frac{a_{n} - a_{1}}{1 + a_{n} a_{1}}$
$[∵ tan ({tan}^{- 1} x) = x, x ϵ R]$

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Similar questions

If $a_{1}, a_{2}, a_{3}, . . ., a_{n}$ is an arithmetic progression with common difference d, then evaluate the following expression.

$t a n [t a n^{- 1} (\frac{d}{1 + a_{1} a_{2}}) + t a n^{- 1} (\frac{d}{1 + a_{2} a_{3}}) + t a n^{- 1} (\frac{d}{1 + a_{3} a_{4}}) + \dots + t a n^{- 1} (\frac{d}{1 + a_{n - 1} a_{n}})]$

Q. If

a_{1}, a_{2}, a_{3}, \dots, a_{n}

are in arithmetic progression with common difference

d,

then evaluate the following expression :

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_{3} a_{4}}) + . . . . . . + {tan}^{- 1} (\frac{d}{1 + a_{n - 1} a_{n}})}

Q. If

a_{1,} a_{2}, a_{3} \dots

are in A.P. with common difference 'd',then

tan {{tan}^{- 1} (\frac{d}{1 + a_{1} a_{2}}) + {tan}^{- 1} (\frac{d}{1 + a_{2} a_{3}}) + \dots + (\frac{d}{1 + a_{n - 1} a_{n}})}

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Q.
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are in A.P. with common difference d, then

t a n [t a n^{- 1} (\frac{d}{1 + a_{1} a_{2}}) + t a n^{- 1} (\frac{d}{1 + a_{2} a_{3}}) + \dots \dots \dots \dots \dots \dots + t a n^{- 1} (\frac{d}{1 + a_{n - 1} a_{n}})] =

Q. If

θ = t a n^{- 1} \frac{d}{1 + a_{1} a_{2}} + t a n^{- 1} \frac{d}{1 + a_{2} a_{3}} + \dots + t a n^{- 1} \frac{d}{1 + a_{n - 1} a_{n}}

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t a n θ =

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