1.

The value of cot[{Sigma_(n=1)^(23){cot^(-1))1+Sigma_(k=1)^(n)2k}] is

Answer»

`23/25`
`25/23`
`23/24`
`24/23`

SOLUTION :Clearly
`COT^(-1)(1+underset(k=1)overset(N)Sigma 2k)=cot^(-1)1+2underset(k=1)overset(n_Sigmak)=cot^(-1)1+n(n+1)`
`=TAN^(-1)(1)/(1+n(n+1)}=(tan^(-1)(n+1)-n)/(1+n(n+1))`
`tan^(-1)(n1)-tan^(-1)n`
`therefore underset(n=1)overset(23)Sigma(tan^(-1)(n+1)-tan^(-1)n)`
`tan^(-1)24-tan^(-1)1`
`tan^(-1)(24-1)/(1+24xx1)=tan^(-1)(23)/(25)=cot^(-1)25/23`
Hence `cot[underset(n=1)overset(23)Sigma{cot^(-1)(1+underset(k=1)overset(n)Sigma 2k)}]=cot^(-1)25/23=25/23`


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