The geometric mean (GM) between two numbers is a number, which, when placed between them, forms with them a geometric progression.
Formula to calculate the geometric mean
Let m be the geometric mean between two numbers a and b, then a, m, b are in GP.
The common ratio is m/a or b/m, such that m/a =b/m ⇒ m2 = ab (m = ).
The Geometric Mean (GM) between a and b is
Let a = 4 and b = 16
Then = = = 8
4, 8, 16 form a GP.
In general, many numbers can be inserted between any two given numbers.
Let us find n numbers G1, G2, G3,....,Gn between two numbers a and b.
Then the sequence a, G1, G2, G3,....,Gn ,b forms a GP having (n+2) terms.
Let a1 = a, a2 = G1, a3 = G2 , ....., an+1= Gn, an+2= b.
Now, an+2 = b
⇒ arn+2-1 = b, where r is the common ratio of this GP
ar(n+1) = b
r(n+1) = b/a
Then G1 = t2 = ar
= a x = a
G2 = t3 = ar2
= a x = a
G3 = t4 = ar3 = a x = a ……….. and so on.
Likewise, we have Gn = tn+1 = arn = a x = a
Relationship between AM and GM
The arithmetic mean between two numbers is their average.
The arithmetic mean (AM) between two numbers is a number, which, when placed between them, forms with them an arithmetic progression.
Let A and G be the AM and GM of two positive real numbers a and b, respectively.
Then A = and = .
Now, A - G = –
= ≥ 0
∴ A – G ≥ 0
⇒ A ≥ G
The AM between two positive real numbers is always greater than or equal to their GM.