The middle term in the expansion of $(5 x - 7 y)^{7}$ .

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Solution

Since n is an odd number, the expansion contains two middle terms.
$(\frac{7 + 1}{2}) t h$ and $(\frac{7 + 3}{2}) t h$ terms are the two middle terms in the expansion of $(5 x - 7 y)^{7}$ .

$T_{4} = T_{3 + 1} = (- 1)^{3} {7_{C}}_{3} (5 x)^{7 - 3} (7 y)^{3} = - {7_{C}}_{3} (5 x)^{4} (7 y)^{3}$

$T_{5} = T_{4} + 1 = (- 1)^{4} . {7_{C}}_{4} (5 x)^{7 - 4} (7 y)^{4} = {7_{C}}_{4} (5 x)^{3} (7 y)^{4}$ .

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