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What is an Arithmetic meansWhat 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) 7,13,19Solution: 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 + $\frac{3(b  a) }{n + 1}$     $A_{n}$ = a + nd = a + $\frac{n(b  a) }{n + 1}$ Examples on inserting arithmetic means between the numbers1) 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. Covid19 has led the world to go through a phenomenal transition . Elearning is the future today. Stay Home , Stay Safe and keep learning!!! From what is an arithmetic means to Home Covid19 has affected physical interactions between people. Don't let it affect your learning.
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