|Class 11 Maths Sequences Series||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, ar2, ar3, ar4 --(ii)
Comparing (i) & (ii), we get a=1 & ar4 = 256 or r = ± 4
For r = 4, we have G1 = ar = 4,
G2 = ar2 = 16,
G3 = ar3 = 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.