If the ratio of the fifth term from the begining to the fifth term from the end in the expansion of $\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{\mathrm{n}}$ is $\sqrt{6}: 1$, then the third term from the beginning is:

JEE Main · 2023 · 06 Apr 2023 · Shift 1 · Binomial Theorem

If the ratio of the fifth term from the begining to the fifth term from the end in the expansion of

(42 + \frac{1}{4 3})^{n}

6 : 1

, then the third term from the beginning is:

Answer: (B)

Writing

a = 2^{1/4}

and

b = 3^{- 1/4}

, the fifth term from the beginning is

T_{5} = (4 n) a^{n - 4} b^{4} = (4 n) 2^{(n - 4) /4} 3^{- 1}

and the fifth term from the end is

T_{n - 3} = (4 n) a^{4} b^{n - 4} = (4 n) 2 \cdot 3^{- (n - 4) /4}

. Their ratio is

\frac{T _{5}}{T _{n - 3}} = 2^{(n - 8) /4} 3^{(n - 8) /4} = 6^{(n - 8) /4}

. Setting this equal to

6 = 6^{1/2}

gives

\frac{n - 8}{4} = \frac{1}{2}

, so

n = 10

. The third term from the beginning is

T_{3} = (2 10) a^{8} b^{2} = 45 \cdot 2^{2} \cdot 3^{- 1/2} = \frac{180}{3} = 603

JEE Main · 2023 · 06 Apr 2023 · Shift 1 · Single correct option