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Find the positive integers n such that the expansion of (4x^2 - 7y^3)^n contains a term of the form cx^2*y^3.

 Nov 9, 2023
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In order for the expansion of (4x2−7y3)n to contain a term of the form cx2y3, there must be a combination of 4x2 and −7y3 in the expansion that gives a product of cx2y3.

Since n is positive, the only way for this to happen is if the coefficient of x2y3 in the expansion of (4x2−7y3)n is nonzero.

The coefficient of x2y3 in the expansion of (4x2−7y3)n can be found using the binomial theorem:

\binom{n}{3}(4x^2)^3(-7y^3)^3 = \binom{n}{3}(-2192x^6y^9) = -2192\binom{n}{3}x^6y^9

For this coefficient to be nonzero, (3n​) must be nonzero. The smallest positive integer n for which (3n​) is nonzero is n=4, so the only positive integer n such that the expansion of (4x2−7y3)n contains a term of the form cx2y3 is 4​.

 Nov 9, 2023

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