Question

The value of the expression $\text{cot}^{-1 \, } \frac{1}{2} + \text{cot}^{-1 \, } \frac{9}{2} + \text{cot}^{-1 \, } \frac{2 5}{2} + \text{cot}^{-1 \, } \frac{4 9}{2} + \text{.......}$ upto $n$ terms is

• A. $tan^{- 1} 2 n$
• B. $\left(tan\right)^{- 1} \left(2 n - 1\right)$
• C. $tan^{- 1} n$
• D. $tan^{- 1} 2 n-tan^{- 1} ⁡ 1$

Answer 1

Answer: (A)

Given expression $=\tan ^{-1} 2+\tan ^{-1} \frac{2}{9}+\tan ^{-1} \frac{2}{25}+\tan ^{-1} \frac{2}{49}+\ldots \ldots .$
General term $=\frac{2}{(2 n-1)^{2}}=\frac{2}{4 n^{2}-4 n+1}=\frac{2}{1+4 n(n-1)}=\frac{2 n-(2 n-2)}{1+2 n(2 n-2)}$
$T_{n}=\tan ^{-1} 2 n-\tan ^{-1}(2 n-2)$
$\therefore $ Sum of the series $=\tan ^{-1} 2-\tan ^{-1} 0+\tan ^{-1} 4-\tan ^{-1} 2+\tan ^{-1} 6-\tan ^{-1} 4+\ldots \tan ^{-1}$
$2 n-\tan ^{-1}(2 n-2)$
$=\tan ^{-1} 2 n-\tan ^{-1} 0=\tan ^{-1} 2 n$