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 M₁,M₂,M₃,M₄,....M_n is Arithmetic Means between numbers

Then according to rule

a,M₁,M₂,M₃,M₄,....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 M₁ first mean)=a+d=a+(b-a)/(n+1)

third term (or M₂ 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,M₁,M₂,M₃,M₄,M₄,10

we have to find 5 arithmetic mean between 2 and 10

here a = 2 and and t₇=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 M₁=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.