Find the positive integer n such that the expansion of (4x^2 - 7y^3)^n contains a term of the form cx^2*y^6.

hg378 Jun 19, 2023

#1**+1 **

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xud33 Jun 19, 2023

#4**+1 **

The term cx^2y^6 will be present in the expansion of (4x^2 - 7y^3)^n if and only if n is divisible by 2 and 3. This is because the term cx^2y^6 can only be created by combining two factors of x^2 and three factors of y^3.

**The smallest positive integer n that satisfies these conditions is n = 6.** This is because 6 is divisible by 2 and 3, and the expansion of (4x^2 - 7y^3)^6 contains the term 4x^2y^6.

Here is a more detailed explanation of why n must be divisible by 2 and 3.

The term cx^2*y^6 can only be created by combining two factors of x^2 and three factors of y^3. This means that there must be at least two instances of x^2 and at least three instances of y^3 in the expansion of (4x^2 - 7y^3)^n.

The expansion of (4x^2 - 7y^3)^n contains a term of x^2 in the first term, which is 4x^2. This term will be present in the expansion regardless of the value of n.

The expansion of (4x^2 - 7y^3)^n contains a term of y^3 in the third term, which is -7y^3. This term will be present in the expansion regardless of the value of n.

For the term cx^2*y^6 to be present in the expansion, there must be at least one other term in the expansion that contains a factor of x^2 and a factor of y^3.

The only way for this to happen is if n is divisible by 2 and 3. If n is not divisible by 2, then there will be no other terms in the expansion that contain a factor of x^2. If n is not divisible by 3, then there will be no other terms in the expansion that contain a factor of y^3.

Therefore, n must be divisible by 2 and 3 in order for the term cx^2*y^6 to be present in the expansion.

Guest Jun 19, 2023