Question

Find the fourth term from the end in the expansion of(x^3/2-2/x^2)^9​

Answer 1

Fourth term from the end in the expansion is $\mathbf{\frac{672}{x^{3}}}$

Step-by-step explanation:

1. Given data

  $\mathbf{\left ( \frac{x^{3}}{2}-\frac{2}{x^{2}} \right )^{9}}$          ...1)

 Here total number of term = 9+1=10

 Fourth term from last = seventh term from starting

 Means we have to find out seventh term of expansion of given equation 1)

2. $\mathbf{(r+1)^{th}}$ term of equation $\mathbf{(a+b)^{n}}$ is

   

   $\mathbf{T_{r+1}=_{r}^{n}\textrm{C}\times a^{n-r}\times b^{r}}$       ...2)

3. With help of equation 2) seventh term of equation 1) can be written as

   $\mathbf{T_{6+1}=_{6}^{9}\textrm{C}\times \left ( \frac{x^{3}}{2} \right )^{9-6}\times \left ( \frac{-2}{x^{2}} \right )^{6}}$      ...3)

4. Where

   $\mathbf{_{6}^{9}\textrm{C}=\frac{9\times 8\times 7}{3\times 2\times 1}=84}$  

5. Again from equation 3)

   $\mathbf{T_{6+1}=84\times \frac{x^{9}}{2^{3}}\times \frac{2^{6}}{x^{12}}}$

   $\mathbf{T_{6+1}=84\times \frac{2^{3}}{x^{3}}}$

   $\mathbf{T_{6+1}= \frac{672}{x^{3}}}$  = This is answer.