Mathematics - Definite Integration Question with Solution | TestHub

$\begin{array}{rl}& I_K=\int 1\cdot {\left(1-x^7\right)}^{K}dx\\ & I_K={{\left(1-x^7\right)}^{K}x|}_0^1+7K{\int }_0^1{\left(1-x^7\right)}^{K-1}{x}^6\cdot xdx\\ & I_K=-7K{\int }_0^1{\left(1-x^7\right)}^{K-1}\left(\left(1-x^7\right)-1\right)dx\\ & I_K=-7K{I}_K+7K{I}_{K-1}\\ & \Rightarrow \frac{I_K}{I_{K+1}}=\frac{7K+8}{7K+7}\end{array}$
$\begin{array}{rl}& r_K=\frac{7K+8}{7\mathrm{K}+7}\\ & {\mathrm{r}}_{\mathrm{K}}-1=\frac{1}{7(\mathrm{K}+1)}\\ & \Rightarrow 7\left({\mathrm{r}}_{\mathrm{K}}-1\right)=\frac{1}{\mathrm{K}+1}\\ & \sum _{\mathrm{K}=1}^{10}(\mathrm{K}+1)=11(6)-1=65\end{array}$