If one A.M A and two GM p and q are to be inserted between two given numbers, then $\dfrac{{{p}^{2}}}{q}+\dfrac{{{q}^{2}}}{p}=?$
Answer
361.2k+ views
Assume a and b be the two numbers. Let p and q be the G.M. and A.M. respectively. Use the formula for G.M. and A.M. and calculate the values of p and q. Use these values of p and q to calculate the required value.
“Complete step-by-step answer:”
Let a and b be the two numbers between which two G.M.’s p and q and one A.M. A are to be inserted. Let us first insert p and q between a and b so that they form a geometric series
$a\ \ \ p\ \ \ q\ \ \ b$
Now the common ratio r between each number must be the same. Therefore, r of this geometric series can be given as
$r=\dfrac{q}{p}$
Also, since $\dfrac{p}{a}=\dfrac{q}{p}=\dfrac{b}{q}=r$ (common ratio between consecutive terms),
we get
$a=\dfrac{{{p}^{2}}}{q}\ \ \ ,\ \ \ \ b=\dfrac{{{q}^{2}}}{p}$
Now, arithmetic mean A between a and b can be given as
$A=\dfrac{a+b}{2}$
Putting values of a and b we get
$\begin{align}
& A=\dfrac{1}{2}\left( \dfrac{{{p}^{2}}}{q}+\dfrac{{{q}^{2}}}{p} \right) \\
& \therefore \dfrac{{{p}^{2}}}{q}+\dfrac{{{q}^{2}}}{p}=2A \\
\end{align}$
Note: The problems related to arithmetic and geometric mean are based on a simple principle that adding any number of A.M’s and G.M’s between two terms will keep the terms “equidistant” from each other. If A.M 's are added then the “arithmetic distance” or difference between two consecutive terms must be constant while adding G.M.'s one should keep “geometric distance” or ratio between consecutive terms must be constant. Majority of the problems in Arithmetic and Geometric progression can be solved using the basic principles. The concept of arithmetic and geometric means is not limited to just two numbers. The arithmetic mean of n numbers can also be calculated by using formula
$AM=\dfrac{{{a}_{1}}+{{a}_{2}}+{{a}_{3}}....+{{a}_{n}}}{n}$
Geometric mean of n numbers can also be given by the formula
$GM=\sqrt[n]{{{a}_{1}}.{{a}_{2}}.{{a}_{3}}...{{a}_{n}}}$
We should be very careful though, if we had to insert one G.M. between a and b then $\sqrt{ab}$ would be that term but since we had to insert two G.M.’s, we cannot use this term.
“Complete step-by-step answer:”
Let a and b be the two numbers between which two G.M.’s p and q and one A.M. A are to be inserted. Let us first insert p and q between a and b so that they form a geometric series
$a\ \ \ p\ \ \ q\ \ \ b$
Now the common ratio r between each number must be the same. Therefore, r of this geometric series can be given as
$r=\dfrac{q}{p}$
Also, since $\dfrac{p}{a}=\dfrac{q}{p}=\dfrac{b}{q}=r$ (common ratio between consecutive terms),
we get
$a=\dfrac{{{p}^{2}}}{q}\ \ \ ,\ \ \ \ b=\dfrac{{{q}^{2}}}{p}$
Now, arithmetic mean A between a and b can be given as
$A=\dfrac{a+b}{2}$
Putting values of a and b we get
$\begin{align}
& A=\dfrac{1}{2}\left( \dfrac{{{p}^{2}}}{q}+\dfrac{{{q}^{2}}}{p} \right) \\
& \therefore \dfrac{{{p}^{2}}}{q}+\dfrac{{{q}^{2}}}{p}=2A \\
\end{align}$
Note: The problems related to arithmetic and geometric mean are based on a simple principle that adding any number of A.M’s and G.M’s between two terms will keep the terms “equidistant” from each other. If A.M 's are added then the “arithmetic distance” or difference between two consecutive terms must be constant while adding G.M.'s one should keep “geometric distance” or ratio between consecutive terms must be constant. Majority of the problems in Arithmetic and Geometric progression can be solved using the basic principles. The concept of arithmetic and geometric means is not limited to just two numbers. The arithmetic mean of n numbers can also be calculated by using formula
$AM=\dfrac{{{a}_{1}}+{{a}_{2}}+{{a}_{3}}....+{{a}_{n}}}{n}$
Geometric mean of n numbers can also be given by the formula
$GM=\sqrt[n]{{{a}_{1}}.{{a}_{2}}.{{a}_{3}}...{{a}_{n}}}$
We should be very careful though, if we had to insert one G.M. between a and b then $\sqrt{ab}$ would be that term but since we had to insert two G.M.’s, we cannot use this term.
Last updated date: 30th Sep 2023
•
Total views: 361.2k
•
Views today: 5.61k
Recently Updated Pages
What is the Full Form of DNA and RNA

What are the Difference Between Acute and Chronic Disease

Difference Between Communicable and Non-Communicable

What is Nutrition Explain Diff Type of Nutrition ?

What is the Function of Digestive Enzymes

What is the Full Form of 1.DPT 2.DDT 3.BCG

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

The poet says Beauty is heard in Can you hear beauty class 6 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is the past tense of read class 10 english CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
