We will discuss about what is arithmetic mean in mathematics?

When given three quantities are in Arithmetic Progression, the middle one is known as the arithmetic mean of the other two.

Examples of Arithmetic Mean:

(i) In the Arithmetic Progression {12, 22, 32}, 22 is the arithmetic mean between 12 and 32.

(ii) In the Arithmetic Progression {7, 9, 11}, 9 is the arithmetic mean between 7 and 11.

(iii) In the Arithmetic Progression {-5, 6, 17}, 6 is the arithmetic mean between -5 and 17.

(iv) In the Arithmetic Progression {-8, -12, -16}, -12 is the arithmetic mean between -8 and -16.

Let m be the arithmetic mean of two given quantities x and y.
Then, x, m, y are in Arithmetic Progression.

Now, m - x = y - m = common difference

⇒ 2m = x + y

⇒ m = \(\frac{x + y}{2}\)

Therefore, the arithmetic mean between any two given quantities is half their sum.

If more than three terms are in Arithmetic Progress, then the terms between the two extremes are called the arithmetic means between the extreme terms.

Examples on arithmetic means between the extreme terms:

(i) In the Arithmetic Progression {3, 7, 11, 15, 19, 23, 27, 31, 35} the terms 7, 11, 15, 19, 23, 27 and 31 are the arithmetic means between the two extreme terms 3 and 35.

(ii) In the Arithmetic Progression {-5, -2, 1, 4, 7, 10, 13, 16, 19} the terms -2, 1, 4, 7, 10, 13 and 16 are the arithmetic means between the two extreme terms -5 and 19.

(iii) In the Arithmetic Progression {85, 80, 75, 70, 65, 60, 55, 50, 45} the terms 80, 75, 70, 65, 60, 55, and 50 are the arithmetic means between the two extreme terms 85 and 45.

**●** **Arithmetic Progression**

**Definition of Arithmetic Progression****General Form of an Arithmetic Progress****Arithmetic Mean****Sum of the First n Terms of an Arithmetic Progression****Sum of the Cubes of First n Natural Numbers****Sum of First n Natural Numbers****Sum of the Squares of First n Natural Numbers****Properties of Arithmetic Progression****Selection of Terms in an Arithmetic Progression****Arithmetic Progression Formulae****Problems on Arithmetic Progression****Problems on Sum of 'n' Terms of Arithmetic Progression**

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