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