Mathematics - Sequence & Series Question with Solution | TestHub

AM-GM on $1,2,4,\dots ,2^{n-1}$ gives $2^n-1>n\cdot 2^{(n-1)/2}$, proving Statement 1. Statement 2 implies $\mathrm{AM}/\mathrm{GM}=\mathrm{GM}/\mathrm{HM}>1$, hence $\mathrm{AM}>\mathrm{GM}$ for distinct positives. General AM-GM on $1,2,4,\dots ,2^{n-1}$ proves Statement 1, but the two-number identity alone does not explain that n-number inequality. The key is B because Statement 2 concerns two numbers and does not by itself explain the n-number inequality.