Question

The term independent of $a$ in the expansion of ${(1+\sqrt{a}+\frac{1}{\sqrt{a}-1})}^{-30}$ is

• A. ${}^{30}{C}_{20}$
• B. $0$
• C. ${}^{30}{C}_{10}$
• D. none of these

Answer 1

The correct option is D none of these

The above equation can be further simplified
$(\sqrt{a}+1+\frac{1}{\sqrt{a}-1}{)}^{-30}$
$=(\frac{a-1+1}{\sqrt{a}-1}{)}^{-30}$
$=(\frac{\sqrt{a}-1}{a}{)}^{30}$
$=(\frac{1}{\sqrt{a}}-\frac{1}{a}{)}^{30}$
Hence there won't be any term independent of $a$.
The rth term of the expansion can be written as
$(-1{)}^r\times {}^{30}{C}_r\times a^{-15-r/2}$
For the rth term to be independent of 'a', $-15-r/2=0$, which gives negative value of 'r', which is not possible. Hence, there is no term independent of 'a'.