Find the 4th term from the end in the expansion

Question:

Find the 4 th term from the end in the expansion of $\left(\frac{4 x}{5}-\frac{5}{2 x}\right)^{8}$.

Solution:

Let Tr+1 be the 4th term from the end of the given expression.

Then,

Tr+1 is (10 − 4 + 1)th term, i.e., 7th term, from the beginning.

Thus, we have:

$T_{7}=T_{6+1}$

$={ }^{9} C_{6}\left(\frac{4 x}{5}\right)^{9-6}\left(\frac{5}{2 x}\right)^{6}$

$=\frac{9 \times 8 \times 7}{3 \times 2}\left(\frac{64}{125} x^{3}\right)\left(\frac{125 \times 125}{64 x^{6}}\right)$

 

$=\frac{10500}{x^{3}}$

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