Question

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

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

Answer 1

Answer: (B)

${\left(1+\sqrt{a}+\frac{1}{\sqrt{a}-1}\right)}^{-30}$
$={\left(\frac{a}{\sqrt{a}-1}\right)}^{-30}$
$={\left(\frac{\sqrt{a}-1}{a}\right)}^{30}$
$=\frac{1}{a^{30}}(1-\sqrt{a}{)}^{30}$
$=\frac{1}{a^{30}}\left\{{}^{30}{C}_0-{}^{30}{C}_1\sqrt{a}+\dots +{}^{30}{C}_{30}(\sqrt{a}{)}^{30}\right\}$
There is no term independent of $a$.