# Arithmetic Mean:

#### What is Arithmetic Mean?

The middle term between the two other terms of an arithmetic series is called the arithmetic mean.

Example: Let an arithmetic series is 2, 4, 6, 8, 10,.......

Here "4" is the arithmetic mean of 2 and 6 similarly, "6" is the arithmetic mean of 4 and 8.

#### The arithmetic mean of two terms "a" and "b":

Let "M" is the arithmetic mean of a and b then it can be written as $$a, M, b,.....$$ Hence, $$M - a = b - M$$ $$2M = a + b$$ $$\bbox[5px,border:1px solid black] { M = \frac{1}{2} \ (a + b) }$$

Example: Find the arithmetic mean of 10 and 20?

Solution: Let a = 10 and b = 20, then arithmetic mean, $$M = \frac{1}{2} \ (a + b)$$ $$M = \frac{1}{2} \ (10 + 20)$$ $$M = \frac{30}{2} = 15$$

#### "n" Arithmetic means between two terms a and b:

Let n arithmetic means between a and b are $$M_1, M_2, M_3.....M_n$$ then it can be written in arithmetic series as $$a, M_1, M_2, M_3.....M_n, b$$ Hence total number of terms in this Arithmetic series will be (n + 2).

Let the common difference of the series is "d" then $$(n + 2)^{th}$$ term, $$n^{th} \ term = a + (n - 1) \ d$$ $$Then \ (n + 2)^{th} \ term$$ $$b = a + (n + 2 - 1) \ d$$ $$b = a + (n + 1) \ d$$ $$b - a = (n + 1) \ d$$ $$d = \frac{b - a}{n + 1}$$ Hence first term of the series, $$M_1 = a + d$$ $$M_1 = a + \frac{b - a}{n + 1}$$ $$M_1 = \frac{an + b}{n + 1}....(1)$$ Now the second term of the series, $$M_2 = a + 2d$$ $$M_2 = a + 2 \times \frac{b - a}{n + 1}$$ $$M_2 = \frac{an + a + 2b - 2a}{n + 1}$$ $$M_2 = \frac{an - a + 2b}{n + 1}$$ $$M_2 = \frac{a (n - 1) + 2b}{n + 1}.....(2)$$ Similarly $$n^{th}$$ term of the series $$M_n = a + nd$$ $$M_n = a + n \times \frac{b - a}{n + 1}$$ $$M_n = \frac{a + bn}{n + 1}$$