
If \[{A_1},{A_2};{G_1},{G_2}\]and \[{H_1},{H_2}\]be two A.M.s, G.M.s and H.M.s between two numbers respectively, then \[\frac{{{G_1}{G_2}}}{{{H_1}{H_2}}} \times \frac{{{H_1} + {H_2}}}{{{A_1} + {A_2}}} = \]
A. \[1\]
B. \[0\]
C. \[2\]
D. \[3\]
Answer
162k+ views
Hint
The total number of data values is n, so multiply all the numbers together and calculate the nth root of the multiplied numbers. The arithmetic mean (AM) and the harmonic mean (HM) are multiplied to create the geometric mean (GM). The product of the two integers' roots provides the geometric mean. We always get to the conclusion that the two numbers in the given series are equal to one another when we take into account all the alternatives.
The arithmetic means of a list of non-negative real numbers are greater than or equal to the geometric means of the same list, according to the AM-GM inequality, also known as the inequality of arithmetic and geometric means. Only a possibility exists if each number on the list is the same for the two numbers.
Formula use:
To find the ratio of two numbers,
H.M of a and b \[ = \frac{{2ab}}{{(a + b)}}\]
G.M of a and b \[ = \sqrt {(ab)} \]
\[AM \times HM{\rm{ }} = {\rm{ }}G{M^2}\]
The sum of n A.M.s is equal to \[n \times \]single A.M.
the product of n G.M.s is equal to single G.M. \[ \times n\]
Complete step-by-step solution
Assume a and b be two numbers.
The sum of n A.M.s is equal to \[n \times \]single A.M.
\[ = > {A_1} + {A_2} = 2 \times (\frac{{a + b}}{2})\]
This equation thus becomes
\[a + b\]
Similarly, the product of n G.M.s is equal to single G.M. \[ \times n\]
The equation becomes,
\[{G_1} \times {G_2} = {(\sqrt {ab} )^2}\]\[ = ab\]
The series of numbers in A.P is
\[\frac{1}{a},\frac{1}{{{H_1}}},\frac{1}{{{H_2}}},\frac{1}{b}\]
\[ = > \frac{1}{{{H_1}}} + \frac{1}{{{H_2}}} = \frac{1}{a} + \frac{1}{b}\]
This then becomes,
\[\frac{{a + b}}{{ab}}\]
\[ = > \frac{{{H_1}{H_2}}}{{{H_1} + {H_2}}} = \frac{{{G_1}{G_2}}}{{{A_1} + {A_2}}}\]
This equation is finally calculated as
\[ = > \frac{{{G_1}{G_2}}}{{{H_1}{H_2}}} = \frac{{{H_1} + {H_2}}}{{{A_1} + {A_2}}} = 1\]
Therefore, the correct option is A.
Note
The arithmetic mean is calculated by adding a set of numbers, dividing by their count, and then taking the result. The Geometric Mean is a unique kind of average in which the numbers are multiplied together and then their square, cube, or other root is taken (for two numbers, three numbers, etc.).
The geometric mean, also known as the geometric mean (GM), while the arithmetic mean is the average of two numbers.
The total number of data values is n, so multiply all the numbers together and calculate the nth root of the multiplied numbers. The arithmetic mean (AM) and the harmonic mean (HM) are multiplied to create the geometric mean (GM). The product of the two integers' roots provides the geometric mean. We always get to the conclusion that the two numbers in the given series are equal to one another when we take into account all the alternatives.
The arithmetic means of a list of non-negative real numbers are greater than or equal to the geometric means of the same list, according to the AM-GM inequality, also known as the inequality of arithmetic and geometric means. Only a possibility exists if each number on the list is the same for the two numbers.
Formula use:
To find the ratio of two numbers,
H.M of a and b \[ = \frac{{2ab}}{{(a + b)}}\]
G.M of a and b \[ = \sqrt {(ab)} \]
\[AM \times HM{\rm{ }} = {\rm{ }}G{M^2}\]
The sum of n A.M.s is equal to \[n \times \]single A.M.
the product of n G.M.s is equal to single G.M. \[ \times n\]
Complete step-by-step solution
Assume a and b be two numbers.
The sum of n A.M.s is equal to \[n \times \]single A.M.
\[ = > {A_1} + {A_2} = 2 \times (\frac{{a + b}}{2})\]
This equation thus becomes
\[a + b\]
Similarly, the product of n G.M.s is equal to single G.M. \[ \times n\]
The equation becomes,
\[{G_1} \times {G_2} = {(\sqrt {ab} )^2}\]\[ = ab\]
The series of numbers in A.P is
\[\frac{1}{a},\frac{1}{{{H_1}}},\frac{1}{{{H_2}}},\frac{1}{b}\]
\[ = > \frac{1}{{{H_1}}} + \frac{1}{{{H_2}}} = \frac{1}{a} + \frac{1}{b}\]
This then becomes,
\[\frac{{a + b}}{{ab}}\]
\[ = > \frac{{{H_1}{H_2}}}{{{H_1} + {H_2}}} = \frac{{{G_1}{G_2}}}{{{A_1} + {A_2}}}\]
This equation is finally calculated as
\[ = > \frac{{{G_1}{G_2}}}{{{H_1}{H_2}}} = \frac{{{H_1} + {H_2}}}{{{A_1} + {A_2}}} = 1\]
Therefore, the correct option is A.
Note
The arithmetic mean is calculated by adding a set of numbers, dividing by their count, and then taking the result. The Geometric Mean is a unique kind of average in which the numbers are multiplied together and then their square, cube, or other root is taken (for two numbers, three numbers, etc.).
The geometric mean, also known as the geometric mean (GM), while the arithmetic mean is the average of two numbers.
Recently Updated Pages
If there are 25 railway stations on a railway line class 11 maths JEE_Main

Minimum area of the circle which touches the parabolas class 11 maths JEE_Main

Which of the following is the empty set A x x is a class 11 maths JEE_Main

The number of ways of selecting two squares on chessboard class 11 maths JEE_Main

Find the points common to the hyperbola 25x2 9y2 2-class-11-maths-JEE_Main

A box contains 6 balls which may be all of different class 11 maths JEE_Main

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JoSAA JEE Main & Advanced 2025 Counselling: Registration Dates, Documents, Fees, Seat Allotment & Cut‑offs

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More
