Question

find the term independent of x in the expansion of (3x^2-1/2x^3)^10
PLEASE SEND COMPLETE SOLUTION!!!!​

Answer 1

The $(r+1)^{th}$ term in the expansion of $\left(3x^2-\dfrac{1}{2x^3}\right)^{10}$ is,

$\displaystyle\longrightarrow T_{r+1}=\,^{10}C_r\,(3x^2)^{10-r}\left(-\dfrac{1}{2x^3}\right)^r$

$\displaystyle\longrightarrow T_{r+1}=(-1)^r\cdot\,^{10}C_r\,3^{10-r}x^{2(10-r)}\cdot\dfrac{1}{2^rx^{3r}}$

$\displaystyle\longrightarrow T_{r+1}=(-1)^r\cdot\,^{10}C_r\,3^{10-r}x^{20-5r}\cdot\dfrac{1}{2^r}$

For getting term independent of $x,$ the exponent of $x$ should be 0.

$\longrightarrow 20-5r=0$

$\longrightarrow r=4$

Hence 5th term is independent of $x$ and is,

$\displaystyle\longrightarrow T_5=(-1)^4\cdot\,^{10}C_4\cdot3^{10-4}\cdot\dfrac{1}{2^4}$

$\displaystyle\longrightarrow\underline{\underline{T_5=\dfrac{76545}{8}}}$