4.02: Binomial expansion and constant term

Find, in descending powers of \(x\), the first three terms in the expansion of \(\left(x^2+\dfrac{1}{2x}\right)^9\). Give each term in its simplest form.

The expansion of \(\left(x^2+\dfrac{1}{2x}\right)^n\), where \(n\) is a positive integer, has a term that is independent of \(x\). Show that \(n\) is divisible by \(3\).