If one G.M. $G$ and two geometric means $p$ and $q$ be inserted between any two given numbers, then ${{G}^{2}}=\left( 2p-q \right)\left( 2q-p \right)$ .
A) True
B) False
Last updated date: 20th Mar 2023
•
Total views: 306.6k
•
Views today: 5.84k
Answer
306.6k+ views
Hint: The given problem is related to the geometric mean of two numbers. Here we will use the formulae related to the insertion of geometric means between two numbers.
Complete step-by-step answer:
Before proceeding with the solution, first, we will understand the concept of the geometric mean.
The geometric mean of a series with $n$ terms is defined as the ${{n}^{th}}$ root of the product of the terms of the series.
For two numbers, the geometric mean is defined as the square root of the product of the two numbers.
If $n$ geometric means are inserted between two numbers, then the series formed as such will be a geometric progression.
Now, coming to the question, it is given that $G$ is the geometric mean of two numbers. So, let the two numbers be $A$ and $B$. Since $G$ is the geometric mean of $A$ and $B$, so $A$, $G$, and $B$ will be in geometric progression.
So, $\dfrac{G}{A}=\dfrac{B}{G}$.
$\Rightarrow {{G}^{2}}=AB$
Now, it is also given that two geometric means $p$ and $q$ are also inserted between the two given numbers $A$ and $B$.
So, $A,p,q$ and $B$ are in geometric progression.
So, $\dfrac{p}{A}=\dfrac{q}{p}=\dfrac{B}{q}$.
$\Rightarrow {{p}^{2}}=Aq....(i)$ and ${{q}^{2}}=Bp.....(ii)$
From equation$(i)$ , we have ${{p}^{2}}=Aq$.
$\Rightarrow A=\dfrac{{{p}^{2}}}{q}$
From equation$(ii)$ , we have ${{q}^{2}}=Bp$.
$\Rightarrow B=\dfrac{{{q}^{2}}}{p}$
So, $AB=\dfrac{{{p}^{2}}}{q}\times \dfrac{{{q}^{2}}}{p}$.
$=pq$
Now, we have ${{G}^{2}}=AB$ and $AB=pq$.
So, ${{G}^{2}}=pq$.
Hence, the statement that ${{G}^{2}}=\left( 2p-q \right)\left( 2q-p \right)$ is false.
Therefore, the answer is option B.
Note: Don’t get confused between arithmetic mean and geometric mean. The geometric mean of a series with $n$ terms is defined as the ${{n}^{th}}$ root of the product of the terms of the series, whereas the arithmetic mean of a series is defined as the average of the series.
Complete step-by-step answer:
Before proceeding with the solution, first, we will understand the concept of the geometric mean.
The geometric mean of a series with $n$ terms is defined as the ${{n}^{th}}$ root of the product of the terms of the series.
For two numbers, the geometric mean is defined as the square root of the product of the two numbers.
If $n$ geometric means are inserted between two numbers, then the series formed as such will be a geometric progression.
Now, coming to the question, it is given that $G$ is the geometric mean of two numbers. So, let the two numbers be $A$ and $B$. Since $G$ is the geometric mean of $A$ and $B$, so $A$, $G$, and $B$ will be in geometric progression.
So, $\dfrac{G}{A}=\dfrac{B}{G}$.
$\Rightarrow {{G}^{2}}=AB$
Now, it is also given that two geometric means $p$ and $q$ are also inserted between the two given numbers $A$ and $B$.
So, $A,p,q$ and $B$ are in geometric progression.
So, $\dfrac{p}{A}=\dfrac{q}{p}=\dfrac{B}{q}$.
$\Rightarrow {{p}^{2}}=Aq....(i)$ and ${{q}^{2}}=Bp.....(ii)$
From equation$(i)$ , we have ${{p}^{2}}=Aq$.
$\Rightarrow A=\dfrac{{{p}^{2}}}{q}$
From equation$(ii)$ , we have ${{q}^{2}}=Bp$.
$\Rightarrow B=\dfrac{{{q}^{2}}}{p}$
So, $AB=\dfrac{{{p}^{2}}}{q}\times \dfrac{{{q}^{2}}}{p}$.
$=pq$
Now, we have ${{G}^{2}}=AB$ and $AB=pq$.
So, ${{G}^{2}}=pq$.
Hence, the statement that ${{G}^{2}}=\left( 2p-q \right)\left( 2q-p \right)$ is false.
Therefore, the answer is option B.
Note: Don’t get confused between arithmetic mean and geometric mean. The geometric mean of a series with $n$ terms is defined as the ${{n}^{th}}$ root of the product of the terms of the series, whereas the arithmetic mean of a series is defined as the average of the series.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
