Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of 2 4 + 1 3 4 n , in the increasing powers of 1 3 4 be 6 4 : 1 . If the sixth term from the beginning is α 3 4 , then α is equal to _______.

Question

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of 2 4 + 1 3 4 n , in the increasing powers of 1 3 4 be 6 4 : 1 . If the sixth term from the beginning is &#945; 3 4 , then &#945; is equal to _______.

Aicharya · Accepted Answer

Given,
The ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of ${(\sqrt[4]{2} + \frac{1}{\sqrt[4]{3}})}^{n}$ in the increasing powers of $\frac{1}{\sqrt[4]{3}}$ be $\sqrt[4]{6} : 1$
Now according to given data we have,
$\frac{T_{5}}{T_{n - 3}} = \frac{{}^{n}C_{4} {(2^{1 / 4})}^{n - 4} {(3^{- 1 / 4})}^{4}}{{}^{n}C_{n - 4} {(2^{1 / 4})}^{4} {(3^{- 1 / 4})}^{n - 4}} = \frac{\sqrt[4]{6}}{1}$
$\Rightarrow 2^{\frac{n - 8}{4}} 3^{\frac{n - 8}{4}} = 6^{1 / 4}$
$\Rightarrow 6^{n - 8} = 6$
$\Rightarrow n - 8 = 1 \Rightarrow n = 9$
So, $T_{6} = {}^{9}C_{5} {(2^{1 / 4})}^{4} {(3^{- 1 / 4})}^{5} = \frac{84}{\sqrt[4]{3}}$
Now comparing with $T_{6} = \frac{α}{\sqrt[4]{3}}$ ,
We get $α = 84$

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion…

Solution