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What is an Arithmetic means

What is an Arithmetic means ?
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

∴ A = $\frac{a + b}{2}$
Thus the required arithmetic mean (A.M) of two numbers 'a' and 'b' is $\frac{a + b}{2}$

Example 1 : Consider the following A.P.
3,8,13,18,23,28,33
Here the first term is 3 and the last term is 33 and there are 5 numbers between the two . So all these five numbers are called 'arithmetic means' between 3 and 33

Example 2 : Consider the A.P. as 4,8,12,16,20
Here the first term is 4 and the last term is 20. There are three terms ( 8,12,and 16) between the two terms. So these three terms are 'arithmetic means' between 4 and 20.

What is an arithmetic means between the following ?

1) 7,13,19
Solution: Arithmetic mean between 7 and 19 is 13.
2) 6,9,12,15,18
Solution: Arithmetic means between 6 and 18 is 9,12 and 15.
3) 'a' is an arithmetic between 6 and 18 .Find 'a'
Solution: 'a' is an A.M between 6 and 18
∴ a = $\frac{6 + 18}{2}$
a = $\frac{24}{2}$
∴ a = 12

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 $A_{1},A_{2},A_{3},...A_{n}$. That means 'n' number of arithmetic means can be inserted between the two numbers 'a' and 'b'.

Let a, $A_{1},A_{2},A_{3},...A_{n}$, b is in A.P.
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
∴ d = $\frac{b - a }{n + 1}$

Thus 'n' arithmetic means between 'a' and 'b' are as follows
$A_{1}$ = a + d = a + $\frac{b - a }{n + 1}$

$A_{2}$ = a + 2d = a + $\frac{2(b - a) }{n + 1}$

$A_{3}$ = a + 3d = a + $\frac{3(b - a) }{n + 1}$

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$A_{n}$ = a + nd = a + $\frac{n(b - a) }{n + 1}$

Examples on inserting arithmetic means between the numbers

1) Insert three arithmetic means between 8 and 26.
Solution: Let three arithmetic numbers inserted will be $A_{1},A_{2} and A_{3}$ between 8 and 26.
∴ 8, $A_{1},A_{2},A_{3}$, 26 are in A.P.Then
a = 8 and b = 26 and n = 5
∴ $a_{n}$ = a +(n -1)d
26 = 8 + 4d
18 = 4d
$\frac{18}{4} = \frac{4d}{4}$
∴ d = 4.5
$A_{1}$ = a + d = 8 + 4.5 = 12.5
$A_{2} = a + 2d = 8 + 2\times$ 4.5 = 17
$A_{3} = a + 3d = 8 + 3\times$ 13.5 = 21.5
Thus the three arithmetic means between 8 and 26 are 12.5, 17 and 21.5.



11th grade math

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