Mathematics - Definite Integration Question with Solution | TestHub

$\begin{array}{rl}& I_n={\int }_0^1{\left(1-x^k\right)}^n\cdot 1dx\\ & I_n={\left(1-x^k\right)}^n\cdot x-nk{\int }_0^1{\left(1-x^k\right)}^{n-1}\cdot x^{k-1}\cdot dx\\ & I_n=nk{\int }_0^1\left[{\left(1-x^k\right)}^n-{\left(1-x^k\right)}^{n-1}\right]dx\\ & I_n=nk{I}_n-nk{I}_n\\ & \frac{I_n}{I_{n-1}}=\frac{nk}{nk+1}\\ & \frac{I_{21}}{I_{20}}=\frac{21k}{1+21k}\\ & =\frac{147}{148}\Rightarrow k=7\end{array}$