Imagine a number, call it n<\/strong>. You multiply it by a number, r<\/strong>, to get nr<\/strong>. Then you multiply it by r<\/strong> again, to get nr2<\/sup><\/strong>.<\/p>\n n nr nr2<\/sup><\/p>\n In the sequence above, the geometric mean is nr<\/strong>, the middle number.<\/p>\n Consider the numerical example<\/p>\n 5 20 80<\/p>\n Unlike the arithmetic mean,<\/a> the geometric mean is not equidistant from the earlier and later numbers. It’s one multiple by r<\/strong> from the first, and one division by r<\/strong> from the last.<\/p>\n In most problems, finding that multiplying number r<\/strong> is the critical idea.<\/p>\n Example:<\/strong> Find the geometric mean between 14 and 42.<\/p>\n Solution: We imagine the sequence<\/p>\n 14 ? 42<\/p>\n At the same time, we can imagine that sequence as<\/p>\n n nr nr2<\/sup><\/p>\n From the first two representations, we put together<\/p>\n 14 14r 14r2<\/sup>=42<\/p>\n From<\/p>\n 14r2<\/sup>=42<\/p>\n we divide both sides by 14, to get<\/p>\n r2<\/sup>=42\/14=3<\/p>\n Finally, square rooting both sides, we arrive at<\/p>\n r=\u00b1√3<\/p>\n Apparently, our geometric mean, known before as 14r<\/span>, turns out to have two possible values: 14√3<\/span>, or else -14√3<\/span>.<\/p>\n I’ll be continuing this topic in future posts. HTH:)<\/p>\n Source:<\/p>\n Travers, Kenneth et al. Using Advanced Algebra<\/em>. Toronto: Doubleday Canada Limited, 1977.<\/p>\n