Solve the following; If ${a^2}$,${b^2}$,${c^2}$ are in A.P., show that $b + c$, $c + a$,$a + b$ are in H.P.
Last updated date: 29th Mar 2023
•
Total views: 308.4k
•
Views today: 2.85k
Answer
308.4k+ views
Hint: First, We should try to convert given terms ${a^2}$, ${b^2}$, ${c^2}$ to the terms somewhat similar to terms what is to be shown i.e. $b + c$,$c + a$,$a + b$ by adding or subtracting some additional terms, see if some factors are made or not.
Given ${a^2},{b^2},{c^2}$ are in A.P.
By adding $(ab + ac + bc)$ to each term of given A.P.
We see that ${a^2} + ab + ac + bc$,${b^2} + ab + ac + bc$,\[{c^2} + ab + ac + bc\] are also in A.P.
Convert each term in factor form
${a^2} + ab + ac + bc = a\left( {a + b} \right) + c\left( {a + b} \right) = \left( {a + c} \right)\left( {a + b} \right)$
${b^2} + ab + ac + bc = b\left( {b + a} \right) + c\left( {a + b} \right) = \left( {a + b} \right)\left( {b + c} \right)$
${c^2} + ac + ab + bc = c\left( {c + a} \right) + b\left( {a + c} \right) = \left( {c + b} \right)\left( {a + c} \right)$
Also we can write terms of above A.P. in another way like
$\left( {a + b} \right)\left( {a + c} \right)$,$\left( {a + b} \right)\left( {b + c} \right)$,$\left( {c + a} \right)\left( {c + b} \right)$ are in A.P.
Now we divide each term by $\left( {a + b} \right)\left( {b + c} \right)\left( {c + a} \right)$
We get$\dfrac{1}{{b + c}}$,$\dfrac{1}{{c + a}}$,$\dfrac{1}{{a + b}}$ are in A.P.
So we can say that$b + c$,$c + a$,$a + b$ are in H.P.
Hence proved
Note: Arithmetic progression (A.P) means a sequence in which each differs from the preceding one by a constant quantity. Harmonic progression means a series when their reciprocal is in arithmetic progression.
Given ${a^2},{b^2},{c^2}$ are in A.P.
By adding $(ab + ac + bc)$ to each term of given A.P.
We see that ${a^2} + ab + ac + bc$,${b^2} + ab + ac + bc$,\[{c^2} + ab + ac + bc\] are also in A.P.
Convert each term in factor form
${a^2} + ab + ac + bc = a\left( {a + b} \right) + c\left( {a + b} \right) = \left( {a + c} \right)\left( {a + b} \right)$
${b^2} + ab + ac + bc = b\left( {b + a} \right) + c\left( {a + b} \right) = \left( {a + b} \right)\left( {b + c} \right)$
${c^2} + ac + ab + bc = c\left( {c + a} \right) + b\left( {a + c} \right) = \left( {c + b} \right)\left( {a + c} \right)$
Also we can write terms of above A.P. in another way like
$\left( {a + b} \right)\left( {a + c} \right)$,$\left( {a + b} \right)\left( {b + c} \right)$,$\left( {c + a} \right)\left( {c + b} \right)$ are in A.P.
Now we divide each term by $\left( {a + b} \right)\left( {b + c} \right)\left( {c + a} \right)$
We get$\dfrac{1}{{b + c}}$,$\dfrac{1}{{c + a}}$,$\dfrac{1}{{a + b}}$ are in A.P.
So we can say that$b + c$,$c + a$,$a + b$ are in H.P.
Hence proved
Note: Arithmetic progression (A.P) means a sequence in which each differs from the preceding one by a constant quantity. Harmonic progression means a series when their reciprocal is in arithmetic progression.
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths 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

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE
