Challenging Binomial Question; O Level A Maths Group Tuition

Question: (Broadrick Sec Prelim Add Math Paper 1 2010, Q8b)

In the expansion of $\displaystyle (x^2-\frac{1}{2x^4})^n$ , in descending powers of $x$ , the seventh term is independent of $x$ . Find the value of $n$ and the value of this term.

Solution:

$\displaystyle\begin{array}{rcl} T_{r+1}&=&{n \choose r}(x^2)^{n-r}(-\frac{1}{2}x^{-4})^r\\ &=& {n\choose r}x^{2n-2r}(-\frac{1}{2})^r (x^{-4r})\\ &=& {n\choose r}(-\frac{1}{2})^r x^{2n-6r} \end{array}$