**What Is Arithmetic Mean**

If three or more than three terms are in A.P., then the numbers lying between first and last term are known as Arithmetic Means between them.i.e.

The A.M. between the two given quantities a and b is

A so that a, A, b are in A.P.

i.e. A – a = b – A

( Rightarrow A=frac{a+b}{2} )

**Note:** A.M. of any n positive numbers a_{1}, a_{2} ……a_{n} is

( A=frac+++…..}{n} )

**n AM’s between two given numbers**

If in between two numbers ‘a’ and ‘b’ we have to insert n AM A_{1}, A_{2}, …..An then a, A_{1}, A_{2}, A_{3}….A_{n}, b will be in A.P. The series consist of (n + 2) terms and the last term is b and first term is a.

a + (n + 2 – 1) d = b

( d=frac{b-a}{n+1} )

A_{1} = a + d, A_{2} = a + 2d, …… A_{n} = a + nd or

A_{n} = b – d

**Note:**

**(i)** Sum of n AM’s inserted between a and b is equal to n times the single AM between a and b i.e.

( sumlimits_{r,=,1}^{n}}=nAtext{ Where }A=frac{a+b}{2} )

**(ii)** between two numbers

( =frac{sum,of,m,AM’s}{sum,of,n,AM’s}=frac{m}{n} )

**Arithmetic Mean Examples**

**Example 1:** If 4 AM’s are inserted between 1/2 and 3 then find 3rd AM.

**Solution. ** Here

( d=frac{3-frac{1}{2}}{4+1}=frac{1}{2} )

∴ A_{3} = a + 3d

( Rightarrow frac{1}{2}+3times frac{1}{2}=2 )

**Example 2: **n AM’s are inserted between 2 and 38. If third AM is 14 then find n.

**Solution. ** Here 2 + 3d = 14 ⇒ d = 4

( therefore 4=frac{38-2}{n+1} )

⇒ 4n + 4 = 36

⇒ n = 8