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  • In <img alt="\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^n" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cleft%28%5Csqrt%5B3%5D%7B2%7D&plus;%5Cfrac%7B1%7D%7B%5Csqrt%5B3%5D%7B3

In \left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^n , if the ratio of 7^{th} term from the beginning to the 7^{th} term from the end is \frac{1}{6}, then n =

Option: 1

7


Option: 2

8


Option: 3

9


Option: 4

None of these


Answers (1)

best_answer

\frac{1}{6}=\frac{{ }^n C_6\left(2^{1 / 3}\right)^{n-6}\left(3^{-1 / 3}\right)^6}{{ }^n C_{n-6}\left(2^{1 / 3}\right)^6\left(3^{-1 / 3}\right)^{n-6}}  or

\begin{aligned} & 6^{-1}=6^{-4} \cdot 6^{n / 3}=6^{n / 3-4} \\ & \therefore \frac{n}{3}-4=-1 \Rightarrow n=9 \text {. } \\ & \end{aligned}

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Rishabh

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