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How do I approach this problem using mathematical induction?

$$\frac{(2n)!}{(n!)^2} \leq 2^{2n}$$

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    You should verify it for $n=1$. Then show that when you assume it's true for $n=k$ it also holds for $n=k+1$. Or just realize the RHS is the sum of the binomial coefficients in the $2n$'th row of Pascal's triangle, and the LHS is one such element.2011-12-06

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