Geometric Mean
The geometric mean of two positive numbers a and b is the number √ab. Therefore, the geometric mean of 2 and 8 is 4. We observe that the three numbers 2,4,8 are consecutive terms of a G. P. This leads to a generalization of the concept of geometric means of two numbers.
Given any two positive numbers a and b, we can insert as many numbers as we like between them to make the resulting sequence in a G. P.
Numerical: Insert three numbers between 1 and 256 so that the resulting sequence is a G. P.
Solution: Let G1, G2, G3 be three numbers between 1 and 256 such that 1, G1, G2, G3, 256 is a G. P. –( i)
We can rewrite this GP as a, ar, ar 2, ar 3, ar 4--( ii) Comparing( i) &( ii), we get a = 1 & ar 4 = 256 or r = ± 4 For r = 4, we have G1 = ar = 4, G2 = ar 2 = 16, G3 = ar 3 = 64 Similarly, for r = – 4, numbers are – 4,16 and – 64.
Hence, we can insert 4, 16, 64 or – 4, 16 and – 64 between 1 and 256 so that the resulting sequences are in G. P.
Relationship between AM & GM
Let A and G be A. M. and G. M. of two given positive real numbers a and b, respectively.
Then AM =( a + b)/ 2 & GM =( a * b) ½
AM ≥ GM