Answer
Verified
351k+ views
Hint: In the given question we are given some terms of arithmetic progression and some terms of geometric progression and some more terms are also given which we need to identify what kind of series it belongs to or is any kind of progression for which we need to clearly use the definition of each terms.
Complete step by step answer:
According to the question, we are given that a, b, c is in AP and we know that in any arithmetic progression the common difference between the consecutive terms is the same. Therefore, we can write as
$\begin{align}
& b-a=c-b \\
& \Rightarrow 2b=c-a \\
& \Rightarrow b=\dfrac{c-a}{2} \\
\end{align}$
Now, similarly we know that in GP the common ratio between the consecutive terms is same therefore, we get,
$\begin{align}
& \dfrac{mb}{a}=\dfrac{c}{mb} \\
& \Rightarrow {{m}^{2}}{{b}^{2}}=ac \\
\end{align}$
Now, making some substitutions in this we get,
$\begin{align}
& {{m}^{2}}bb=ac \\
& \Rightarrow {{m}^{2}}b\left( \dfrac{c-a}{2} \right)=ac \\
& \Rightarrow {{m}^{2}}b=\dfrac{2ac}{c-a} \\
\end{align}$
And this clearly shows the given three terms are in HP as they satisfy the relation of harmonic series.
Therefore, a, ${{m}^{2}}b$, c is in HP.
Note: In such a type of question, we need to be careful with the definition of each type of progression or series involved. Also, we need to be aware of all the main terms involved and proceed directly as asked by the definition and always try to write the simplified answer and work step wise step in order to reduce chances of mistakes.
Complete step by step answer:
According to the question, we are given that a, b, c is in AP and we know that in any arithmetic progression the common difference between the consecutive terms is the same. Therefore, we can write as
$\begin{align}
& b-a=c-b \\
& \Rightarrow 2b=c-a \\
& \Rightarrow b=\dfrac{c-a}{2} \\
\end{align}$
Now, similarly we know that in GP the common ratio between the consecutive terms is same therefore, we get,
$\begin{align}
& \dfrac{mb}{a}=\dfrac{c}{mb} \\
& \Rightarrow {{m}^{2}}{{b}^{2}}=ac \\
\end{align}$
Now, making some substitutions in this we get,
$\begin{align}
& {{m}^{2}}bb=ac \\
& \Rightarrow {{m}^{2}}b\left( \dfrac{c-a}{2} \right)=ac \\
& \Rightarrow {{m}^{2}}b=\dfrac{2ac}{c-a} \\
\end{align}$
And this clearly shows the given three terms are in HP as they satisfy the relation of harmonic series.
Therefore, a, ${{m}^{2}}b$, c is in HP.
Note: In such a type of question, we need to be careful with the definition of each type of progression or series involved. Also, we need to be aware of all the main terms involved and proceed directly as asked by the definition and always try to write the simplified answer and work step wise step in order to reduce chances of mistakes.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Why Are Noble Gases NonReactive class 11 chemistry CBSE
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
At which age domestication of animals started A Neolithic class 11 social science CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Write a letter to the principal requesting him to grant class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE