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If the coefficient of x⁷ in (ax² + (1)/(bx))¹¹ is equal to the coefficient of x⁻⁷ in (ax - (1)/(bx²))⁷, then the constants a and b must obey which relation?

1. a - b = 1
2. a + b = 1
3. a/b = 1
4. ab = 1

Correct answer: ab = 1

Solution

In (a x^2 + 1/(bx))^11 the x^7 term is at k=5: C(11,5) a^6 b^-5. In the second bracket (power 11) the x^-7 term is at j=6: C(11,6) a^5 b^-6. Since C(11,5)=C(11,6), a^6/b^5 = a^5/b^6, giving ab=1.

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