Question

Sum of first n terms in the following series

${cot}^{-1}$3 + ${cot}^{-1}$7 + ${cot}^{-1}$13 + ${cot}^{-1}$21 + .........n terms.is given by

• A. ${tan}^{-1}$( $\frac{n}{n+2}$)
• B. ${cot}^{-1}$( $\frac{n+2}{n}$)
• C. ${tan}^{-1}$(n+1) - ${tan}^{-1}$ 1
• D. All of these

Answer 1

The correct option is D

All of these

Let S = 3 + 7 + 13 + 21 + ...... + $T_{\pi }$ ⇒ $T_{\pi }$ = $n^2$ + n + 1.

Let $T_r$ = ${cot}^{-1}$( $r^2$ + r + 1) = ${tan}^{-1}$(r + 1) - ${tan}^{-1}$r.

Put r = 1, 2,........, n and add, we get the required sum

${tan}^{-1}$(n+1) - ${tan}^{-1}$1 = ${tan}^{-1}$( $\frac{n}{n+2}$) =${cot}^{-1}$ ($\frac{n+2}{n}$)