# Arithmetic Means(A.M.’s)

Arithmetic Means is a value that comes between two number and series become an Arithmetic series (make a series an Arithmetic Series).

There are mainly two type of Arithmetic Means

1. Single Arithmetic Means

2. n Arithmetic Means

Single Arithmetic means-: Any number is known as single arithmetic means between two numbers if all three term together become arithmetic series.

For example if a and b are two given number and M is Arithmetic Means then series a,M,b must be a arithmetic series

Ex 1,3,5 in this series 3 is Arithmetic Means between 1 and 5

2. n Arithmetic Means between two numbers-: series of n numbers is known as n arithmetic means between two numbers if and only if given number and n number between given number are in arithmetic progression.

example if a,M1,M2,M3.........Mn,b are in arithmetic progression then M1,M2,M3.........Mn are n Arithmetic Means between a and b.

## How to find single Arithmetic Means between two numbers

If a and b two numbers and M is Arithmetic Means between numbers

Then according to rule

M-a=b-M

2M=a+b

M= (a+b)/2

Find single A.M between 1 and 7

M=(1+7)/2=4

4 is single Arithmetic Mean between 1 and 7

### How to find n Arithmetic Means between two numbers

If a and b two numbers and M_{1},M_{2},M_{3},M_{4},....M_{n} is Arithmetic Means between numbers

Then according to rule

a,M_{1},M_{2},M_{3},M_{4},....M_{n},b will be in arithmetic progression

and b = (n+2)^{th} term of arithmetic progression.

t_{n+2}=a+(n+2-1)d

t_{n+2}=a+(n+1)d

t_{n+2} is b so

b=a+(n+1)d

d=(b-a)/(n+1)

Now we have common difference d so

second term (or M_{1} first mean)=a+d=a+(b-a)/(n+1)

third term (or M_{2} second mean)=a+2d=a+2(b-a)/(n+1)

like above we can find all arithmetic mean

Example:

Find 5 Arithmetic Means between 2 and 10

here series will become a,M_{1},M_{2},M_{3},M_{4},M_{4},10

we have to find 5 arithmetic mean between 2 and 10

here
a = 2 and and t_{7}=10

so as per formulas t_{n}=a+(n-1)d

10=2+(7-1)d

10-2=6d

d=8/6

now we know common difference so we can find all five terms or means

first mean between 2 and 10 is M_{1}=second term

second term=first term + common difference=2+8/6=20/6

third term=second term+ common difference=20/6+8/6

according to this we can find all five Arithmetic Means between 2 and 10.