Question

x, y, z are three G.M. between $ 6,54. $ then what is the value of z?

Answer 1

Hint: In GM, the ratio of terms of the geometric progression of the consecutive terms remains the same. Here, we will use the formula for the nth term of the series and with the given values of first and place three unknown terms between the two and simplify for the required term that will be the fourth term.

Complete step-by-step answer:
Let us assume the first five GM’s be given as –
 $ {G_1},{G_2},{G_3},{G_4},{G_5} $
So, the given series can be represented as –
 $ 6,x,y,z,54 $
Above series suggest that the first term of the given series will be $ a = 6 $
Fifth term $ = 54 $
Place the standard formula in the above expression –
 $ a{r^4} = 54 $
Place the known values in the above expression –
 $ 6{r^4} = 54 $
Make the required term “r” the subject –
 $ {r^4} = \dfrac{{54}}{6} $
Remove common factors from the numerator and the denominator –
 $ {r^4} = 9 $
The term can be expressed in the form of power and exponent –
 $ {r^4} = {\left( {\sqrt 3 } \right)^4} $
Common powers from both the sides of the equation cancels each other.
 $ r = \sqrt 3 $
Now, fourth GM can be given by $ {G_4} = a{r^3} $
Place the known values in the above expression –
 $ z = 6{\left( {\sqrt 3 } \right)^3} $
Simplify the above expression by removing the common factors from the numerator and the denominator.
 $ z = 18\sqrt 3 $
The value of $ z = 18\sqrt 3 $
So, the correct answer is “ $ z = 18\sqrt 3 $ ”.

Note: Power and exponent can be defined as the ways of expressing the same factors in the short form. Be good in multiples and get the factors of the terms. If the terms on both sides of the equation are equal and bases are same then powers are equal and vice-versa.