## Task B-4.3.

Determine the values of the real parameter $a$, if the coefficient of the linear term in the expansion of the binomial $\left(x+\frac{1}{a x^{2}}\right)^{7}$ is equal to $\frac{7}{3}$?
## Solution.

The general term in the expansion of the binomial is given by the expression $\binom{7}{k} x^{7-k}\left(\frac{1}{a x^{2}}\right)^{k}$

or, after simplification, $\binom{7}{k} x^{7-k}\left(\frac{1}{a x^{2}}\right)^{k}=\binom{7}{k} x^{7-k-2 k} a^{-k}$.

The linear term is the one for which the exponent of $x$ is equal to 1, i.e.

$$
\begin{aligned}
& 7-3 k=1 \\
& k=2
\end{aligned}
$$

Now, $\binom{7}{2} a^{-2}=\frac{7}{3}$, and from this $a^{2}=9$, i.e., $a= \pm 3$.

Note: If the student writes only $a=3$, they lose 1 point.