6 6 Geometric Mean – Class 11TH Math

6.6 Geometric Mean (G.M.)

$G$ is a G.M. between $a$ and $b$ if $a, G, b$ are in G.P.:

G = \pm\sqrt{ab}

$G_1, \ldots, G_n$ are $n$ G.M.s between $a$ and $b$ if $a, G_1, \ldots, G_n, b$ form a G.P.

Insert three G.M.s between $2$ and $\dfrac{1}{2}$ .

Solution

$2, G_1, G_2, G_3, \dfrac{1}{2}$ is a G.P. with $n = 5$ :

\dfrac{1}{2} = 2r^4 \implies r^4 = \dfrac{1}{4} \implies r = \pm\dfrac{1}{\sqrt{2}}

(also complex roots $\pm\dfrac{i}{\sqrt{2}}$ ). For real $r = \dfrac{1}{\sqrt{2}}$ :

G_1 = \sqrt{2},\quad G_2 = 1,\quad G_3 = \dfrac{1}{\sqrt{2}}

For $r = -\dfrac{1}{\sqrt{2}}$ : $G_1 = -\sqrt{2}$ , $G_2 = 1$ , $G_3 = -\dfrac{1}{\sqrt{2}}$ .