Geometric Mean between Two Numbers
In a geometric progression containing three numbers; say: a, b, c; b is said to be the geometric mean between two numbers a and c. Numbers a and c can be regarded as the extremes.
By definition of Geometric Progression, we have,
$\dfrac{ \text{b}}{\text{a}} \ = \ \dfrac{ \text{c}}{\text{b}}$
$\therefore \text{b}^2 \ = \ \text{ac}$
Geometric Mean between any Two Numbers
Let us consider a geometric progression containing N number of means as follows:
$\text{a, m1, m2, m3, m4, ....., m}_\text{N}\text{, b}$
This progression, being geometric, the terms can also be generalized as
$\text{a, ar, ar}^\text{2}\text{, ar}^\text{3}\text{, ar}^\text{3} \text{, ....,}\text{ ar}^\text{N} \text{, b}$
where $\text{m1 = ar, m2 = ar}^\text{2}\text{, m3 = ar}^\text{3}\text{, m4 = ar}^\text{4}\text{, and m}_\text{N}\text{ = ar}^\text{N}$
where m1, m2, m3, …, mN are the geometric mean between given numbers a and b. Hence, the Nth mean between any two terms of a geometric progression can be calculated by the formula $$\text{m}_\text{N} \text{ = ar}^\text{N}$$In the following section, we shall look into how to obtain an expression for the common difference (d). For this, we know, the formula for the general term of a geometric progression is:
$\text{t}_\text{n} \text{ = ar}^\text{n-1}$
where n refers to the total number of terms. Clearly, the total number of terms in the above progression is n = (N+2). So, the last term of the progression 'b' is defined as
$\text{t}_\text{n} \text{ = ar}^\text{(N+2)-1}$
$\text{b} \text{ = ar}^\text{N+1}$
$\therefore \text{r =} \left ( \dfrac{ \text{b} }{ \text{a} } \right )^{ \frac{1}{\text{N + 1}}}$
Hence, the formula for the general term of a geometric mean of any arithmetic progression can be further explained as:
$\text{m}_\text{N} \text{ = ar}^\text{N}$
$\text{m}_\text{N} \text{ = a}\left ( \dfrac{ \text{b} }{ \text{a} } \right )^{ \frac{1}{\text{N + 1}} \times \text{N}}$
$\text{m}_\text{N} \text{ = a}\left ( \dfrac{ \text{b} }{ \text{a} } \right )^{ \frac{\text{N}}{\text{N + 1}}}$