Insertion of n arithmetic mean in given two numbers

Mar 02, 2022, 16:45 IST

Insertion of n arithmetic mean in given two numbers

What is an arithmetic means ?

Definition :-When three numbers a, a and b are in a.p., then a is called the arithmetic mean of numbers 'a' and 'b'.

Given that, a, a , b is in a.p. then
as they are in a.p. so their common difference will be constant.
A – a = d and b – a = d
∴ a – a = b - a
2a = a + b

Thus the required arthemetic mean (A. M) of two number ‘a’ and ‘b’ is

in the given two numbers, any number of arithmetic means can be inserted between them.

Let the two numbers be 'a' and 'b' and the arithmetic means inserted will be a1,a2,a3,...ana1,a2,a3,...an. That means 'n' number of arithmetic means can be inserted between the two numbers 'a' and 'b'.

Formula description :-
Here b is the (n + 2)th term
So, b = a + [(n + 2) – 1]d
b = a + (n + 1)d
b – a = (n + 1)d

Thus 'n' arthmetic means between ‘a’ and ‘b’ are as follows

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.

Also Check

• General Terms of Arithmetic Progression
• Sum to n Terms of Arithmetic Progression