Arithmetic Mean between Two Numbers
In an arithmetic progression containing three numbers; say: a, b, c; b is said to be the arithmetic mean between two numbers a and c. Numbers a and c can be regarded as the extremes.
By definition of Arithmetic Progression, we have,
$$\text{b - a = c - b}$$
$$\text{2b = a + c}$$
$$\therefore \text{b = } \dfrac{ \text{a + c}}{2}$$
Arithmetic Means between any Two Numbers
Let us consider an arithmetic progression containing N number of means as follows:
$$\text{a, m1, m2, m3, m4, .....,}\text{m}_\text{N} \text{, b}$$
This progression, being arithmetic, the terms can also be generalized as
$$\text{a, a+d, a+2d, a+3d, a+4d, ....., a+Nd, b}$$
where $\text{m1 = a + d, m2 = a + 2d, m3 = a +3d and } \text{m}_\text{N} \text{= a + Nd}$
where m1, m2, m3, ... , mN are the arithmetic mean between given numbers a and b. Hence, the Nth mean between any two terms of an arithmetic progression can be calculated by the fomula $$\text{m}_\text{N} \ = \ \text{a + Nd}$$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 general term of an arithmetic progression is:
$$\text{t}_\text{n} \text{ = a + (n-1)d}$$
where n refers to total number of terms. Clearly, total number of terms in the above progression is n = (N+2). So, the last term of the progression 'b' is defined as
$$\text{b = a + \{(N+2) - 1\}d}$$
$$\text{b = a + (N+1)d}$$
$$\text{b - a = (N+1)d}$$
$$\therefore \text{d =} \dfrac{ \text{b - a } }{\text{N+1}}$$
Hence, the formula for general term of an arithmetic mean of any arithmetic progression can be further explained as:
$$\text{m}_\text{N} \ = \ \text{a + Nd}$$
$$\text{m}_\text{N} \text{ = a + N} \left ( \dfrac{ \text{b - a } }{\text{N+1}} \right )$$
Solved Examples
EXAMPLE 1: Find the artihmetic mean between 2 and 6.
Solution:
Here,
$\text{a = 2, c = 6, b = ?}$
The arithmetic mean between 2 and 6 is given by $\text{b = }\dfrac{\text{a+b}}{2}$
$\text{= }\dfrac{2 + 6}{2}$
$\text{= }\dfrac{8}{2}$
$\text{= 4}$