The value of ( _ n=1 ^ 23 ^ -1 (1+ _ k=1 ^n 2 k ) ) is - Vidyadip

MCQ

The value of $\cot \left(\sum_{n=1}^{23} \cot ^{-1}\left(1+\sum_{k=1}^n 2 k\right)\right)$ is

A
$\frac{23}{25}$
✓
$\frac{25}{23}$
C
$\frac{23}{24}$
D
$\frac{24}{23}$

✓

Answer

Correct option: B.

$\frac{25}{23}$

b
$\cot \sum_{n=1}^{23} \cot ^{-1}(1+2+4+6+\ldots . .+2 n) $

$\cot \sum \cot ^{-1}(1+n(n+1)) $

$\cot \sum \tan ^{-1} \frac{(n+1)-n}{1+n(n+1)} $

$\cot \sum_{n=1}^{23}\left(\tan ^{-1}(n+1)-\tan ^{-1} n\right) $

$\cot \left(\tan ^{-1} 24-\tan ^{-1} 1\right) $

$\cot \left(\tan ^{-1} \frac{24-1}{1+24}\right) $

$\cot \left(\cot ^{-1} \frac{25}{23}\right)=\frac{25}{23}$

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