Question:
The coefficient of $x^{-6}$, in the expansion of $\left(\frac{4 x}{5}+\frac{5}{2 x^2}\right)^9$, is______
The coefficient of $x^{-6}$, in the expansion of $\left(\frac{4 x}{5}+\frac{5}{2 x^2}\right)^9$, is______
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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.
Updated On: July 22, 2025
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Correct Answer: 5040
Approach Solution - 1
Now,
Coefficient of i.e.
So, Coefficient of
So, the correct answer is 5040.
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Approach Solution -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 \( 5040 \).
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.