[tex]\begin{gathered} To\text{ calculate geometric mean of any two numbers} \\ \text{Let the geometric mean of numbers a and c be k} \\ \text{then the sequence is a , k, c} \\ \text{common ratio r = }\frac{k}{a} \\ \text{also, common ratio r can also be written as r = }\frac{c}{k} \\ \text{equating the two common ratios} \\ \frac{k}{a}=\frac{c}{k} \\ k^2=ac \\ k=\sqrt[]{ac} \\ \text{Hence the geometric mean of two numbers a and c is }\sqrt[]{ac} \end{gathered}[/tex]
Now , to find the geometric mean of 9 and 16
[tex]\begin{gathered} a=9,\text{ c= 16} \\ \text{Geometric mean, k =}\sqrt[]{ac\text{ }} \\ =\sqrt[]{9\text{ x 16}} \\ =\sqrt[]{144} \\ =12 \end{gathered}[/tex]
