Question

In the binomial expansion of ${(\sqrt{y}+\frac{1}{2\sqrt[4]{4y}})}^8$, the terms in the expansion in which the power of $y$ is a natural number are:

• A. $T_1,T_5$
• B. $T_1,T_2,T_3,T_4$
• C. $T_2,T_4$
• D. $T_1,T_3,T_5$

Answer 1

The correct option is A $T_1,T_5$

$T_{r+1}={}^n{C}_r{a}^{n-r}{b}^r$
Applying to the above question, we get
$T_{r+1}={}^n{C}_r{y}^{4-\frac{r}{2}}{2}^{-r}{y}^{\frac{r}{4}}$
Power of $y$ $=4-\frac{3r}{4}$
For the power of $y$ to be natural $4-\frac{3r}{4}>0$ and $4-\frac{3r}{4}$ should not be a fraction.
Hence, $r$ must be a multiple of $4$ or $0$.
These conditions are met for $r=0$ and $r=4$.
For $r=8$ , the power of $y$ would be negative
Hence the terms are $T_{0+1}$ and $T_{4+1}$ i.e. $T_1,T_5$