
06672978
j
Varga, Marek
Michali\v{c}ka, Peter
Arithmetic mean and geometric mean.
Acta Math. Nitriensia 2, No. 2, 4348 (2016).
2016
Constantine the Philosopher University in Nitra, Faculty of Natural Sciences, Department of Mathematics, Nitra
EN
I20
I40
F60
arithmetic mean
geometric mean
proof
differential calculus
doi:10.17846/AMN.2016.2.2.4348
Summary: In mathematics we define several types of means. Probably the most known are the arithmetic and geometric means. If we are given $n$ nonnegative numbers $x_1$, $x_2ยง, \dots, $x_n$; then the number $A_n=\frac{x_1+x_2+\dots +x_n}{n}$ we call the arithmetic mean and the number $G_n=\sqrt[n]{x_1x_2\cdots x_n}$ we call the geometric mean of the numbers given. In the first part of the paper with the use of functions' their of more variables we will show that for each natural number $n$ $A_n\ge G_n$ applies. In the second part we will try to show the same, however, without using the differential calculus.