Binomial Theorem
Ratio of Terms from Beginning and End
nta_pyq_2023_apr
Grade 11
Question:
If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $\left(\sqrt[4]{2}+\dfrac{1}{\sqrt[4]{3}}\right)^n$ is $\sqrt{6}:1$, then the third term from the beginning is:
$30\sqrt{2}$
$30\sqrt{3}$
$60\sqrt{2}$
$60\sqrt{3}$
Step-by-Step Solution
Key Concept: $T_5=\binom{n}{4}(\sqrt[4]{2})^{n-4}(\sqrt[4]{3})^{-4}$ and $T'_5=\binom{n}{4}(\sqrt[4]{3})^{-(n-4)}(\sqrt[4]{2})^4$. Their ratio gives $(\sqrt{6})^{(n-8)/2}=\sqrt{6}$.
$n=10$. $T_3=\binom{10}{2}\cdot 2^2\cdot 3^{-1/2}=45\cdot\frac{4}{\sqrt{3}}=60\sqrt{3}$.
Correct Answer: 4