Question

: If \[{a^{\dfrac{1}{x}}} = {b^{\dfrac{1}{y}}} = {c^{\dfrac{1}{z}}}\] and \[a,b,c\] are in geometric progression , then \[x,y,z\] are in
A.AP
B.GP
C.HP
D.None of these

Answer 1

Hint: Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio. This progression is also known as a geometric sequence of numbers that follow a pattern.

Complete step-by-step answer:
General form of a GP
\[a,ar,a{r^2},a{r^3},...,a{r^n}\]
where \[a\] is the first term
\[r\] is the common ratio
\[{r^n}\] is the last term
If the common ratio is:
Negative: the result will alternate between positive and negative.
Greater than \[1\] : there will be an exponential growth towards infinity (positive).
Less than \[ - 1\] : there will be an exponential growth towards infinity (positive and negative).
Between \[1\] and \[ - 1\]: there will be an exponential decay towards zero.
Zero: the result will remain at zero
When three quantities are in GP, the middle one is called the geometric mean of the other two.
A sequence of numbers is called an Arithmetic progression if the difference between any two consecutive terms is always the same. In simple terms, it means that the next number in the series is calculated by adding a fixed number to the previous number in the series
We are given \[{a^{\dfrac{1}{x}}} = {b^{\dfrac{1}{y}}} = {c^{\dfrac{1}{z}}}\]
Let \[{a^{\dfrac{1}{x}}} = {b^{\dfrac{1}{y}}} = {c^{\dfrac{1}{z}}} = \lambda \]
Therefore \[a = {\lambda ^x},b = {\lambda ^y},c = {\lambda ^z}\]
Now we are given that \[a,b,c\] are in geometric progression .
Therefore \[{b^2} = ac\]
Therefore we get ,
\[{\left( {{\lambda ^y}} \right)^2} = \left( {{\lambda ^x}} \right)\left( {{\lambda ^z}} \right)\]
Which becomes
\[{\lambda ^{2y}} = {\lambda ^{x + z}}\]
Since the base is the same. Therefore on comparing the powers we get ,
\[2y = x + z\]
Therefore we get \[x,y,z\] are in AP .
So, the correct answer is “Option A”.

Note: Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio. A sequence of numbers is called an Arithmetic progression if the difference between any two consecutive terms is always the same.