Question

How many terms in the expansion of ${(\frac{3}{a}+\frac{a}{3})}^{10}$ have positive power of $a$, how many have negative powers of $a$?

Answer 1

Since $T_6=T_{r+1}⟹r=5$

$T_6{=}^{10}{C}_5\left(3/a{)}^5\right(a/3{)}^5{=}^{10}{C}_5(a{)}^0$

Since $T_6$ is independent of $a$,

thus half terms will have positive power while other half will have negative power.

So, $5$ terms will have positive power while other $5$ terms will have negative power with $T_6$ being independent of $a$.