Mathematics - Binomial Theorem Question with Solution | TestHub

Let $t_r=\frac{{}^n{C}_{r-1}}{{}^n{C}_r+{}^n{C}_{r-1}}=\frac{{}^n{C}_{r-1}}{{}^{n+1}{C}_r}=\frac{{}^n{C}_{r-1}}{\frac{n+1}{r}{}^n{C}_{r-1}}$ $\therefore l_r=\frac{r}{n+1}$ Now, $S={\sum }_{r=0}^n{\left\{l_r\right\}}^3\Rightarrow S={\sum }_{r=0}^n\frac{r^3}{(n+1{)}^3}=\frac{1}{(n+1{)}^3}{\sum }_{r=0}^n{r}^3$ $\Rightarrow S=\frac{1}{(n+1{)}^3}{\left\{\frac{n(n+1)}{2}\right\}}^2\Rightarrow S=\frac{n^2}{4(n+1)}$ Now, $S=\frac{25}{24}$ (given) which is only possible for 5