In the binomial ((2^ 1 / 3 +3^ -1 / 3 )^ n ), if the ratio of the...

(\mathrm{T}_{\mathrm{r}+1}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}a^{\mathrm{n}-\mathrm{r}} \cdot \mathrm{b}^{\mathrm{r}}) where (\mathrm{a}=2^{1 / 3}) and (\mathrm{b}=3^{-1 / 3}) (\mathrm{T}_{7}) from beginning (={ }^n \mathrm{C}_{6} a^{\mathrm{n}-6} \mathrm{ b}^6) and (\mathrm{T}_{7}) from end (={ }^{\mathrm{n}} \mathrm{C}_{6} \quad \mathrm{ b}^{\mathrm{n}-6} \quad \mathrm{a}^6) $\begin{aligned}& \Rightarrow \frac{\mathrm{a}^{\mathrm{n}-12}}{\mathrm{ b}^{\mathrm{n}-12}}=\frac{1}{6} \Rightarrow2^{\frac{\mathrm{n}-12}{3}} \cdot3^{\frac{\mathrm{n}-12}{3}}=6^{-1} \\ & \Rightarrow \mathrm{n}-12=-3 \Rightarrow \mathrm{n}=9 \end{aligned}$)