Question

If $co{t}^{-1}\left(ɑ\right)=co{t}^{-1}2+co{t}^{-1}8+co{t}^{-1}18+co{t}^{-1}32+\dots \dots ..$ upto $100$ terms, then $ɑ$ is:

• A. $1.03$
• B. $1.00$
• C. $1.01$
• D. $1.02$

Answer 1

The correct option is C

$1.01$

Find the value of $\alpha$:

Let us consider R.H.S $=co{t}^{-1}2+co{t}^{-1}8+co{t}^{-1}18+co{t}^{-1}32+\dots \dots ..$

$$\begin{gathered} =\sum _{n=1}^{100}co{t}^{-1}2n^2 \\ =\sum _{n=1}^{100}ta{n}^{-1}\left(\frac{2}{4n^2}\right) \\ =\sum _{n=1}^{100}ta{n}^{-1}\frac{(2n+1)–(2n–1)}{1+(2n+1)(2n–1)} \\ =\sum _{n=1}^{100}ta{n}^{-1}(2n+1)–ta{n}^{-1}(2n–1) \\ =ta{n}^{-1}201–ta{n}^{-1}1 \\ =ta{n}^{-1}\left(\frac{200}{202}\right) \\ =co{t}^{-1}\left(\frac{101}{100}\right) \end{gathered}$$

$\Rightarrow$ $co{t}^{-1}\left(\alpha \right)=co{t}^{-1}\left(\frac{101}{100}\right)$

$\mathbf{\therefore }\mathit{\alpha }\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{01}$

Hence, Option ‘C’ is Correct.