AGM inequality with binomial expansion (Q1879397)

An ingenious use of the binomial theorem leads to the identity \[ A_n-G_n= { {1}\over{ n}}\sum_{k=2}^n{ n\choose k} A_{n-1}^{(n-k)/n}\bigl(x_n^{1/n}-A_{n-1}^{1/n}\bigr)+ x_n^{1/n}\bigl(A_{n-1}^{1/n}-G_{n-1}^{1/n}\bigr) \] where \(A_k= (x_1 + \cdots +x_k)/k\), and \(G_k= (x_1\ldots x_k)^{1/k},\, 1\leq k\leq n\). The basic inequality \(A_n\geq G_n\) and the extension of Rado \(A_n-G_n\geq {(n-1)\over n}(A_{n-1}-G_{n-1})\) follow in sharpened forms. The two converse inequalities mentioned need more careful consideration.