Question

The coefficient of $x^{-6}$, in the expansion of $\left(\frac{4 x}{5}+\frac{5}{2 x^2}\right)^9$, is______

Hint:

In binomial expansions, identify the power of \( x \) in the general term and solve for the value of \( r \) that gives the desired exponent. Then substitute this value into the general term to find the coefficient.

Answer 1

Correct Answer: 5040

${\left(\frac{4x}{5}+\frac{5}{2x^2}\right)}^9$
Now, $T_{r+1}={}^9{C}_r\cdot {\left(\frac{4x}{5}\right)}^{9-r}{\left(\frac{5}{2x^2}\right)}^r$
$={}^9{C}_r\cdot {\left(\frac{4}{5}\right)}^{9-r}{\left(\frac{5}{2}\right)}^r\cdot x^{9-3r}$
Coefficient of $x^{-6}$ i.e. $9-3r=-6\Rightarrow r=5$
So, Coefficient of $x^{-6}={}^9{C}_5{\left(\frac{4}{5}\right)}^4\cdot {\left(\frac{5}{2}\right)}^5=5040$
So, the correct answer is 5040.

Answer 2

We are given the expansion of: \[ \left( \frac{4x}{5} + \frac{5}{2x^2} \right)^9. \] To find the coefficient of \( x^{-6} \), we use the general term in the binomial expansion: \[ T_r = \binom{9}{r} \left( \frac{4x}{5} \right)^{9-r} \left( \frac{5}{2x^2} \right)^r. \] Step 1: Simplifying the general term: \[ T_r = \binom{9}{r} \left( \frac{4x}{5} \right)^{9-r} \left( \frac{5}{2x^2} \right)^r = \binom{9}{r} \left( \frac{4^{9-r}}{5^{9-r}} \right) x^{9-r} \left( \frac{5^r}{2^r x^{2r}} \right). \] Combining the terms: \[ T_r = \binom{9}{r} \frac{4^{9-r} \cdot 5^r}{5^{9-r} \cdot 2^r} x^{9-r-2r} = \binom{9}{r} \frac{4^{9-r} \cdot 5^r}{5^9 \cdot 2^r} x^{9-3r}. \] Step 2: For the coefficient of \( x^{-6} \), set the exponent of \( x \) equal to \( -6 \): \[ 9 - 3r = -6 \quad \Rightarrow \quad 3r = 15 \quad \Rightarrow \quad r = 5. \] Step 3: Substitute \( r = 5 \) into the general term: \[ T_5 = \binom{9}{5} \frac{4^{9-5} \cdot 5^5}{5^9 \cdot 2^5} x^{-6}. \] Now, calculate the coefficient: \[ \binom{9}{5} = 126, \quad 4^4 = 256, \quad 5^5 = 3125, \quad 5^9 = 1953125, \quad 2^5 = 32. \] Thus, the coefficient is: \[ \text{Coefficient} = 126 \times \frac{256 \times 3125}{1953125 \times 32} = 5040. \] Therefore, the coefficient of \( x^{-6} \) is \( 5040 \).