# Thread: If there exists a non zero term independent of x in the expansion of ..., then n...?

1. ## If there exists a non zero term independent of x in the expansion of ..., then n...?

If there exists a non zero term independent of x in the expansion of [ x^2 - ((2)/(x)^3)]^n , then n can't be:

A) 7
B) 12
C) 9
D) 15

More than one options may be correct.

2. Sorry, I did not notice: "...then n can't be"

${\left( {{x^2} - \frac{2}{{{x^3}}}} \right)^n} = \frac{1}{{{x^{3n}}}} {\left( {{x^5} - 2} \right)^n} = \frac{1}{{{x^{3n}}}}\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}{c}}k \\n \\\end{array} } \right)} {x^{5\left( {n - k} \right)}}{\left( { - 2} \right)^k}.$

For your problem

$\frac{{{x^{5\left( {n - k} \right)}}}}{{{x^{3n}}}} = 1 \Leftrightarrow {x^{5\left( {n - k} \right)}} = {x^{3n}} \Leftrightarrow 5\left( {n - k} \right) = 3n \Leftrightarrow$

$\Leftrightarrow 5k = 2n \Leftrightarrow k = \frac{2}{5}n \Rightarrow n = \left\{ {0;{\text{ }}5;{\text{ }}10;{\text{ }}15;{\text{ }} \ldots ;{\text{ }}5m} \right\}.$

So, right answers are:
A) 7
B) 12
C) 9

3. The general term is $(-1)^kC_n^k(x^2)^{n-k}\left(\frac{2}{x^3}\right)^k=(-1)^kC_n^k2^kx^{2n-5k}$

If the term is independent of x then $2n-5k=0\Rightarrow k=\frac{2n}{5}$

But k is integer, so the only option for n is 15.