Geometric Mean (G.M.) | Quantitative Aptitude for CA Foundation PDF Download

If a, G, b are in G. P., then G is called the geometric mean between a and b.

If three numbers are in G. P., the middle one is called the geometric mean between the other two.

If a, G₁, G₂, ..., G_n, b are in G. P.,

then G₁, G₂, ..., G_n  are called n  G. M.'s between a and b.

The geometric mean of n numbers is defined as the n^th root of their product.

Thus if a₁, a₂, ..., a_n are n numbers, then their

G. M. = (a₁, a₂, ... a_n)^1/n

Let G be the G. M. between a and b, then a, G, b are in G. P.

or, G² =ab

or G = √ab

∴ 

Given any two positive numbers a and b, any number of geometric means can be inserted  a₁, a₂, a₃ ..., a_n be n geometric means between a and b.

then, a₁, a₂, a₃ ..., a_n ,b is a G.P.

Thus, b being the (n + 2)^th term, we have

b = arⁿ⁺¹

or, rⁿ⁺¹ = b/a

or, 

Hence, 

.......     .......

......    .......

Further we can show that the product of these n  G. M.'s is equal to n^th power of the single geometric mean between a and b.

Multiplying a₁, a₂, ... a_n, we have

= (single G. M. between a and b)ⁿ

Example 1. Find the G. M. between 3/2 and 27/2.

Solution: We know that if a is the G. M. between a and b, then

G = √ab

Example 2. Insert three geometric means between 1 and 256.

Solution: Let G₁, G₂, G₃, be the three geometric means between 1 and 256.

Then 1, G₁, G₂, G₃, 256 are in G. P.

If r be the common ratio, then t₅ = 256

i.e, ar⁴ = 256 = 1

r⁴ = 256

or, r² = 16

or r =  ± 4

When r = 4, G₁ = 1. 4 = 4, G₂ = 1. (4)² = 16 and G₃ = 1. (4)³ = 64

When r = – 4, G₁ = – 4, G₂ = (1) (–4)² = 16 and G₃ = (1) (–4)³ = –64

∴ G.M. between 1 and 256 are 4, 16, 64, or, – 4, 16, –64.

Example 3. If 4, 36, 324 are in G. P. insert two more numbers in this progression so thatit again forms a G. P.

Solution:  G. M. between 4 and 

 G. M. between 36 and   

If we introduce 12 between 4 and 36 and 108 betwen 36 and 324, the numbers 4, 12, 36, 108, 324 form a G. P. 

∴ The two new numbers inserted are 12 and 108.

Example 4. Find the value of n such that   may be the geometric mean between a and b.

Solution:  If x be G. M. between a and b, then

x = a^1/2.b^1/2