Question

x>0, n>=1, prove that 1 + x + x² + .....+ x²ⁿ >= (2n+1)xⁿ

Answer 1

$1+x+x^2+...+x^{2n}=\left( 1+x^{2n} \right) +\left( x+x^{2n-1} \right) +...+\left( x^{n-1}+x^{n+1} \right) +x^n=\\=\sum_{k=0}^{n-1}{\left( x^k+x^{2n-k} \right)}+x^n\geqslant \sum_{k=0}^{n-1}{2\sqrt{x^k\cdot x^{2n-k}}}+x^n=2nx^n+x^n=\left( 2n+1 \right) x^n$