If a_1, a_2, a_3, , a_n is an A.P. with common difference d , the… - Vidyadip

MCQ

If $a_1, a_2, a_3, \ldots, a_n$ is an A.P. with common difference d , then
$\tan \left[\tan ^{-1}\left(\frac{d}{1+a_1 a_2}\right)+\tan ^{-1}\left(\frac{d}{1+a_2 a_3}\right)\right.$ $\left.+\ldots+\tan ^{-1}\left(\frac{d}{1+a_{n-1} a_n}\right)\right]=$

A
$\left(\frac{(n-1) d}{a_1+a_n}\right)$
✓
$\left(\frac{(n-1) d}{1+a_1 a_n}\right)$
C
$\left(\frac{n d}{1+a_1 a_n}\right)$
D
$\left(\frac{a_n-a_1}{a_n+a_1}\right)$

✓

Answer

Correct option: B.

$\left(\frac{(n-1) d}{1+a_1 a_n}\right)$

(B) $\tan ^{-1}\left(\frac{d}{1+a_1 a_2}\right)+\tan ^{-1}\left(\frac{d}{1+a_2 a_3}\right)$ $+\ldots \ldots \ldots+\tan ^{-1}\left(\frac{d}{1+a_{n-1} a_n}\right)$
$=\tan ^{-1}\left(\frac{a_2-a_1}{1+a_1 a_2}\right)+\tan ^{-1}\left(\frac{a_3-a_2}{1+a_2 a_3}\right)$ $+\ldots \ldots .+\tan ^{-1}\left(\frac{a_n-a_{n-1}}{1+a_{n-1} a_n}\right)$
$=\left(\tan ^{-1} a_2-\tan ^{-1} a_1\right)+\left(\tan ^{-1} a_3-\tan ^{-1} a_2\right)$ $+\ldots \ldots+\left(\tan ^{-1} a_n-\tan ^{-1} a_{n-1}\right)$
$=\tan ^{-1} a_n-\tan ^{-1} a_1=\tan ^{-1}\left(\frac{a_n-a_1}{1+a_n a_1}\right)$
$=\tan ^{-1}\left(\frac{(n-1) d}{1+a_1 a_n}\right)$

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