If $$tan$$−$$1$$⁡$$11+2+$$$$tan$$−$$1$$⁡$$11+2.3+$$$$tan$$−$$1$$⁡$$11+3.4+$$…$$+$$$$tan$$−$$1$$⁡$$11+n(n+1)=$$$$tan$$−$$1$$⁡x, then x is equal to

Question

If $$tan$$−$$1$$⁡$$11+2+$$$$tan$$−$$1$$⁡$$11+2.3+$$$$tan$$−$$1$$⁡$$11+3.4+$$…$$+$$$$tan$$−$$1$$⁡$$11+n(n+1)=$$$$tan$$−$$1$$⁡x, then x is equal to

Infinity Learn · Accepted Answer

We have, L.H.S. $$=$$$$tan$$−$$1$$⁡$$2$$−$$11+2.1+$$$$tan$$−$$1$$⁡$$3$$−$$21+3.2+$$$$tan$$−$$1$$⁡$$4$$−$$31+4.3+$$,…,$$+$$$$tan$$−$$1$$⁡$$n+1$$¯−$$n1+(n+1)n$$ $$=$$$$tan$$−$$1$$⁡$$2$$−$$tan$$−$$1$$⁡$$1+$$$$tan$$−$$1$$⁡$$3$$−$$tan$$−$$1$$⁡$$2+$$$$tan$$−$$1$$⁡$$4$$−$$tan$$−$$1$$⁡$$3+$$,…,$$+$$$$tan$$−$$1$$⁡(n+1)−$$tan$$−$$1$$⁡n $$=$$$$tan$$−$$1$$⁡(n+1)−$$tan$$−$$1$$⁡$$1=$$$$tan$$−$$1$$⁡(n+1)−$$11+(n+1)1$$ $$=$$$$tan$$−$$1$$⁡$$nn+2=$$$$tan$$−$$1$$⁡x (given) ∴$$x=nn+2$$

If $\tan^{- 1} \frac{1}{1 + 2} + \tan^{- 1} \frac{1}{1 + 2 .3} + \tan^{- 1} \frac{1}{1 + 3 .4} + \dots + \tan^{- 1} \frac{1}{1 + n (n + 1)} = \tan^{- 1} x$ , then x is equal to

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If tan−1⁡11+2+tan−1⁡11+2.3+tan−1⁡11+3.4+…+tan−1⁡11+n(n+1)=tan−1⁡x, then x is equal to

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

If $\tan^{- 1} \frac{1}{1 + 2} + \tan^{- 1} \frac{1}{1 + 2 .3} + \tan^{- 1} \frac{1}{1 + 3 .4} + \dots + \tan^{- 1} \frac{1}{1 + n (n + 1)} = \tan^{- 1} x$ , then x is equal to