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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, G_{1}, G_{2}, ..., G_{n}, b are in G. P.,

then G_{1}, G_{2}, ..., 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 a_{1}, a_{2}, ..., a_{n} are n numbers, then their

G. M. = (a_{1}, a_{2}, ... a_{n})^{1/n}

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

or, G^{2} =ab

or G = âˆšab

âˆ´

Given any two positive numbers a and b, any number of geometric means can be inserted a_{1}, a_{2}, a_{3} ..., a_{n }be n geometric means between a and b.

then, a_{1}, a_{2}, a_{3} ..., a_{n },b is a G.P.

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

b = ar^{n+1}

or, r^{n+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 a_{1}, a_{2}, ... 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 = âˆšab

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

**Solution: **Let G_{1}, G_{2}, G_{3}, be the three geometric means between 1 and 256.

Then 1, G_{1}, G_{2}, G_{3}, 256 are in G. P.

If r be the common ratio, then t_{5} = 256

i.e, ar^{4} = 256 = 1

r^{4} = 256

or, r^{2} = 16

or r = Â± 4

When r = 4, G_{1} = 1. 4 = 4, G_{2} = 1. (4)^{2} = 16 and G_{3 }= 1. (4)^{3} = 64

When r = â€“ 4, G_{1} = â€“ 4, G_{2} = (1) (â€“4)^{2} = 16 and G_{3} = (1) (â€“4)^{3} = â€“64

âˆ´ G.M. between 1 and 256 are 4, 16, 64, or, â€“ 4, 16, â€“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.

âˆ´ 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}

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