Show how geometric means say, G₁, G₂, …, Gn can be inserted between two numbers ‘a’ and ‘b’ in a geometric progression (G.P)?

Expert Solution

Show how geometric means say, G₁, G₂, …, Gn can be inserted between two numbers ‘a’ and ‘b’ in a geometric progression (G.P)?

Sol.

Let common ratio = r

We proceed to insert geometric means like G₁, G₂, …, Gn between ‘a’ and ‘b’ in an geometric progression (A.P) as follows:

So, we proceed to insert geometric means like G₁, G₂, … , Gn between ‘a’ and ‘b’ , where

$∴ G_{1} = a r = a . {(\frac{b}{a})}^{\frac{1}{(n + 1)}}$ , $G_{2} = a r^{2} = a . {(\frac{b}{a})}^{\frac{2}{(n + 1)}}$ $G_{2} = a r^{2} = a . {(\frac{b}{a})}^{\frac{2}{(n + 1)}}$

We proceed to insert geometric means like G₁, G₂, … , Gn between ‘a’ and ‘b’ , where

$∴ G_{1} = a r = a . {(\frac{b}{a})}^{\frac{1}{(n + 1)}}$ , $G_{2} = a r^{2} = a . {(\frac{b}{a})}^{\frac{2}{(n + 1)}}$ $G_{2} = a r^{2} = a . {(\frac{b}{a})}^{\frac{2}{(n + 1)}}$

Shams answered 2 years ago

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