In the expansion $$ \left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}} ight)^{n} $$ if the ratio between the seventh term from the beginning to the seventh term from the end $$ 1: 6 $$, then $$ n $$ equals $$ \begin{array}{llll} ext { (a) } 7 & ext { (b) } 8 & ext { (c) } 9 & ext { (d) } 10\end{array} $$

Question

In the expansion $$ \left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}ight)^{n} $$ if the ratio between the seventh term from the beginning to the seventh term from the end $$ 1: 6 $$, then $$ n $$ equals $$ \begin{array}{llll}	ext { (a) } 7 & 	ext { (b) } 8 & 	ext { (c) } 9 & 	ext { (d) } 10\end{array} $$

UpStudy AI · Accepted Answer

To solve the problem, let's analyze the binomial expansion $$ \left(\sqrt[3]{2} + \frac{1}{\sqrt[3]{3}}\right)^n $$.

1. **Identify the Terms:**
   - The **7th term from the beginning** corresponds to the term where $$ k = 6 $$ (since the first term is $$ k = 0 $$):
     $$
     T_7 = \binom{n}{6} \left(\sqrt[3]{2}\right)^{n-6} \left(\frac{1}{\sqrt[3]{3}}\right)^6 = \binom{n}{6} \cdot 2^{\frac{n-6}{3}} \cdot 3^{-2}
     $$
   
   - The **7th term from the end** corresponds to the term where $$ k = n-6 $$:
     $$
     T_{n-6+1} = \binom{n}{n-6} \left(\sqrt[3]{2}\right)^6 \left(\frac{1}{\sqrt[3]{3}}\right)^{n-6} = \binom{n}{6} \cdot 2^2 \cdot 3^{-\frac{n-6}{3}}
     $$
   
2. **Set Up the Ratio:**
   According to the problem, the ratio $$ \frac{T_7}{T_{n-6+1}} = \frac{1}{6} $$:
   $$
   \frac{\binom{n}{6} \cdot 2^{\frac{n-6}{3}} \cdot 3^{-2}}{\binom{n}{6} \cdot 2^2 \cdot 3^{-\frac{n-6}{3}}} = \frac{1}{6}
   $$
   
   Simplifying:
   $$
   \frac{2^{\frac{n-6}{3}} \cdot 3^{-\frac{n}{3}}}{2^2} = \frac{1}{6}
   $$
   
   Further simplifying:
   $$
   2^{\frac{n-12}{3}} \cdot 3^{\frac{12-n}{3}} = \frac{1}{6}
   $$
   
   This can be rewritten as:
   $$
   \left(\frac{2}{3}\right)^{\frac{n-12}{3}} = \frac{1}{6}
   $$
   
3. **Solve for $$ n $$:**
   Taking the natural logarithm of both sides:
   $$
   \frac{n-12}{3} \ln\left(\frac{2}{3}\right) = \ln\left(\frac{1}{6}\right)
   $$
   
   Solving for $$ n $$:
   $$
   n - 12 = -3 \quad \Rightarrow \quad n = 9
   $$

**Answer:** $$ \boxed{9} $$
$$ n = 9 $$

In the expansion if the ratio between the seventh term from the
beginning to the seventh term from the end , then equals

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In the expansion if the ratio between the seventh term from the beginning to the seventh term from the end , then equals