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Let a, b, c, d be four consecutive binomial coefficients in the expansion of (1 + x)ⁿ where n is a natural number and x > 0. Determine the sign of the expression [b/(b+c)]² - ac/((a+b)(c+d)).

1. < 0
2. > 0
3. = 0
4. Cannot be determined

Correct answer: > 0

Solution

Substituting a=C(n,r), b=C(n,r+1), c=C(n,r+2), d=C(n,r+3) and simplifying using ratio identities shows the expression is always strictly positive for x > 0 and natural n, because the left term exceeds the right term by an AM-GM type inequality on the consecutive coefficient ratios.

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