Find the middle terms 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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