Arithmetic Mean (AM) Formula-Sum of n Arithmetic Means between Two Numbers a & b, Solved Examples

📢 Master Arithmetic Means of an Arithmetic Progression (AP) – Class 11 Mathematics 📚

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The Arithmetic Mean (AM) is a crucial concept in Arithmetic Progression (AP), widely used in mathematics, statistics, physics, and real-world applications. This topic is essential for solving sequence-based problems in Class 11, Board Exams, and JEE Mains/Advanced.

What is Arithmetic Mean?

The arithmetic mean between two numbers is defined to be the sum of numbers divided by the quantity of numbers. Likewise, in a collection, the arithmetic mean is defined to be the sum of the collection of numbers divided by the number of numbers present in the collection.

Another term that is closely associated with arithmetic mean is arithmetic progression. While arithmetic mean is abbreviated as A.M., arithmetic progression is abbreviated as A.P. If three terms are in A.P., then the middle term is called the arithmetic mean (A.M.) between the other two.

Arithmetic Mean (A) Between Two Numbers

If a, b, c are in A.P. then b = (a+c)/2 is the A.M. of a and c.

Let A.M be arithmetic between two numbers a, b, then

A.M. = (a + b)/2

In order to find the arithmetic mean between a and b, we just need to divide the sum of terms by 2. Hence, in the same manner, mathematically, it can be said that if a₁, a₂, ……, a_n are n numbers, then the arithmetic mean of these numbers is

A.M = (a1 + a2 + ….. + an)/n

Hence, if in particular we wish to find the arithmetic mean between 2a and 2b then it is simply a+b. 

Insert n Arithmetic Means Between Two Numbers a, b

The n numbers A₁, A₂, ……, A_n are said to be A.M.’s between the numbers a and b if a, A₁, A₂, ……, A_n, b are in A.P. If d is the common difference of this A.P., then

⇒ b = a + (n + 2 – 1)d

⇒ d = (b–a)/(n+1).

⇒ A₁ = a + (b–a)/(n+1)

⇒ A₂ = a + 2(b–a)/(n+1)

…..

…..

⇒ A_n = a + n(b–a)/(n+1)

Sum of n Arithmetic Means Between Two Numbers a, b

Let S be sum of n Arithmetic Means Between Two Numbers a, b

S = A₁ +A₂, …… + A_n

S = [a + (b–a)/(n+1)] + [a + 2(b–a)/(n+1)] +……….+ [a + n(b–a)/(n+1)]

S = n (a + b)/2

S = n A.M.

Hence sum of n Arithmetic Means between two numbers is equal to n times the arithmetic mean between two numbers.

Solved Examples of Arithmetic Means

Example 1 :-Insert three arithmetic means between 8 and 26.

Solution :-
Let three arthmetic number inserted will be A1, A2 and A3 between 8 and 26.

26 = 8 + 4d
18 = 4d

∴ d = 4.5

A1 = a + d = 8 + 4.5 = 12.5
A2 = a + 2d = 8 + 2 × 4.5 = 17
A3 = a + 3d = 8 + 3 × 13.5 = 21.5

Thus the three arthmetic means between 8 and 26 are 12.5, 17 and 21.5.

Example 2 :-Insert 6 number 3 and 24 such that the resulting sequence is and A.P.

Solution :-
Let A1, A2, A3, A4, A5 and A6 be six number between 3 and 24 such that
3, A1, A2, A3, A4, A5, A6, 24 are in A.P. Here, a = 3, b = 24, n = 8.
Therefore, 24 = 3 + (8 – 1)d, so that d = 3.
Thus,A1 = a + d = 3 + 3 = 6;
A2 = a + 2d = 3 + 2 × 3 = 9;
A3 = a + 3d = 3 + 3 × 3 = 12;
A4 = a + 4d = 3 + 4 × 3 = 15;
A5 = a + 5d = 3 + 5 × 3 = 18;
A6 = a + 6d = 3 + 6 × 3 = 21.

Hence, six numbers between 3 and 24 are 6, 9, 12, 15, 18 and 21.

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