For any two numbers a and b, we can insert a number A between them such that a,A, b are in arithmetic progression. Here, A is called the arithmetic mean of a and b. Since a,A,b are in arithmetic progression, we have
A-a = b-A
2A = a+b
A = (a+b)/2.
Example:
Find the arithmetic mean of 16 and 30.
Solution:
Here a=16 and b=30.
Arithmetic mean = (a+b)/2 = (16+30)/2 = 46/2 = 23.
Geometric mean:
For any two numbers a and b, we can insert a number G between them such that a, G, b are in geometric progression. Here, G is called the arithmetic mean of a and b. Since a,G,b are in geometric progression, we have
G/a = b/G
G2 = ab
G = √ab
Example:
Find the geometric mean of 9 and 16.
Solution:
Here a=9 and b=16.
Geometric mean = √ab=√9*16=√144 = 12.
Relation between arithmetic and geometric means:
Let A and G be the arithmetic and geometric means respectively of two numbers a and b.
Then
A = (a+b)/2
G= √ab.
A-G = (a+b)/2 – √ab
= (a+b-2√ab)/2
= (√a-√b)2/2,
which is always non negative. (≥0).
So, A-G ≥0
i.e. A ≥ G.
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