We know that for any \(n\) positive numbers \({x_1},\,\,{x_2},\,\,{x_3},.......\,{x_n}\), their Arithmetic Mean is more than or equal to their Geometric Mean.

\(\Rightarrow\frac{{{x_1} + {x_2} + {x_3} + .....{x_n}}}{n} \ge {({x_1}{x_2}{x_3}.....{x_n})^{1/n}}\)

This concept can be applied in many problems.

Example: Find the least value of 

\(\left( {\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}} \right)\) given that a, b, c and d are positive numbers.

Solution: Since product of \(\frac{a}{b},\frac{b}{c},\frac{c}{d},\frac{d}{a}\) is constant.

\[\frac{{\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}}}{4} \ge {\left[ {\frac{a}{b}.\frac{b}{c}.\frac{c}{d}.\frac{d}{a}} \right]^{1/4}} \ge 1\]

\[ \Rightarrow \left[ {\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}} \right] \ge 4\]

Example: If \(a, b , c, d, e\) and \(f\) are positive numbers then find the maximum and minimum values of \(G/H\), where \(G = 2a + 3b + 4c\) and

\(H = 5a + 6b + 7c\)

Solution: If \(a\) is very large and \(b, c\) are negligible then \(G/H\) will tend to \(2/5\), similarly If \(b\) is very large and \(a, c\) are negligible then \(G/H\) will tend to \(3/6\), similarly If \(c\) is very large and \(b, a\) are negligible then \(G/H\) will tend to \(4/7\), hence range of \(G/H\) is \((2/5 , 4/7)\)

Example: If \(2a = 3b = 4c\) then find the minimum value of \(a + b + c\). Also find the minimum value of  \(a + b + c\) if \(a, b, c\) are natural.

Solution: From given information,

\(a:b:c = \frac{1}{2}:\frac{1}{3}:\frac{1}{4}\) or \(a : b : c :: 6 : 4 : 3\), hence sum of \(a, b, c\) will be \(13x\) i.e. minimum value of 

\(a + b + c\) can have any value depending on \(x\) may be minus of infinity but if \(a, b, c\) are natural numbers then least value possible for \(x\) is 1 and least value of \(a + b + c\) will be 13.