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

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Algebra Questions

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