Question

If G1,G2,G3,…….Gn. are n G.M’s between two numbers a and b then G1.G2.G3…..Gn=(ab)^n/k where k/4=?

Answer 1

Given : G1,G2,G3,…….Gn. are n G.M’s between two numbers a and b

G1.G2.G3…..Gn=(ab)^n/k

To Find : k/4

Solution:

G1,G2,G3,…….Gn. are n G.M’s between two numbers a and b

GP formed is

a  , G1, G2, G3,…….Gn , b

Number =  n + 2

r  is the common ratio  

b = arⁿ⁺¹     ( n + 2 -  1=  n + 1)

=> rⁿ⁺¹   = b/a

G1 = ar

G2  = ar²

Gn  = arⁿ

G1.G2.G3…..Gn  = ar.ar².................. arⁿ

= aⁿ.r⁽¹⁺²⁺     ⁺ⁿ)

1 + 2 + ....+ n = n(n + 1)/2

rⁿ⁺¹   = b/a

= aⁿ (b/a)^(n.2)

= (ab)^(n/2)

comparing with

(ab)^n/k  

k = 2

k/4 = 2/4  = 1/2

Value of k = 1/2

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