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&Sigma;&kappa; \sqrt{{n \choose k}} <\sqrt{2^{n-1}n^3}, from k=1 to n

Source: Spanish Mathematical Olympiad 1987 P2

August 2, 2018

binomial coefficientsinequalitiespositive integers

Problem Statement

Show that for each natural number

n > 1

1 \cdot \sqrt{{n \choose 1}}+ 2 \cdot \sqrt{{n \choose 2}}+...+n \cdot \sqrt{{n \choose n}} <\sqrt{2^{n-1}n^3}