UPSEE Solved Model Paper 6

We can solve this problem by using Geometric mean between two given numbers.
Geometric mean between two given numbers :
Let a and b be two given positive numbers and G be the Geometric Mean between them. Then a, G, b are in a G.P.
Thus,

G

a

=

b

G

(or) G2 = ab
(or) G = √ab (Since G > 0)
Inserting n-Geometric means between two given numbers :
Let G1,G2, …., Gn be the n geometric means between two given numbers a and b. Then, a, G1,G2, …., Gn, b are in G.P
Now, b = (n+2)th terms of G.P.
= arn+1 ; where r is the common ratio
b = arn+1
rn+1 =

b

a

(or) r = (

b

a

)

1

n+1

and G1 = ar = a(

b

a

)

1

n+1

G2 = ar2 = a(

b

a

)

2

n+1

...
Gn = arn = a(

b

a

)

n

n+1

y using this definition we can find the two geometric means between 1 and 64.
The two geometric means between 1 and 64.
1, G1,G2, 64 are in G.P
b = arn+1 ; here , n = 2
b = ar2+1 = ar3 ;
here, a = 1 , b = 64
64 = 1×r3
r3 = 43
∴ r = 4
G1 = ar = 1 × 4 = 4
G2 = ar2 = 1 × 42 = 16
Therefore, the two geometric means between 1 and 64 are 4 & 16.