In the expansion of (3√2 + 1/3√3)^n, n ∈ N. If the ratio of 15^(th) term from the beginning to the 15^(th) term from the end

Question

In the expansion of \((\sqrt[3]{2} + \frac{1}{\sqrt[3]{3}}), n \in N.\) If the ratio of \(15^{th}\) term from the beginning to the \(15^{th}\) term from the end is \(\frac{1}{6}\), then find the value of \({}^n C_3\)

Answer 1

Answer is: 2300 

In the expansion of \((a + b) ^ n\)  

\(15 ^ {th}\) term from beginning: \(T_{15} = {}^n C_{14} a^{n-14} b^{14}\)  

\(15 ^ {th}\) term from the end: \(T'_{15} = {}^n C_{14} b^{n-14} a^{14}\)  

\(\therefore \frac{T_{15}}{T'_{15}} = \frac{1}{6}\)  

\(\Rightarrow\frac{a^{n-14} b^{14}}{b^{n-14} a^{14}} = \frac{1}{6}\)  

\(\Rightarrow \left(\frac{a}{b}\right)^{n-28} = \frac{1}{6}\)