1963 AHSME Problems/Problem 9

Problem 9

In the expansion of $\left(a-\dfrac{1}{\sqrt{a}}\right)^7$ the coefficient of $a^{-\dfrac{1}{2}}$ is:

$\textbf{(A)}\ -7 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ -21 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 35$


Solution

By the Binomial Theorem, each term of the expansion is $\binom{7}{n}(a)^{7-n}(\frac{-1}{\sqrt{a}})^n$.

We want the exponent of $a$ to be $-\frac{1}{2}$, so \[(7-n)-\frac{1}{2}n=-\frac{1}{2}\] \[-\frac{3}{2}n = -\frac{15}{2}\] \[n = 5\]

If $n=5$, then the corresponding term is \[\binom{7}{5}(a)^{2}(\frac{-1}{\sqrt{a}})^5\] \[-21a^{-\frac{1}{2}}\]

The answer is $\boxed{\textbf{(C)}}$.

See Also

1963 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
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