Question

If \[\left( {\dfrac{{b + c - a}}{a}} \right)\], \[\left( {\dfrac{{c + a - b}}{b}} \right)\], \[\left( {\dfrac{{a + b - c}}{c}} \right)\] are in AP, then a, b, c are in
A. G.P.
B. A.P.
C. H.P.
D. A.G.P.

Answer 1

Hint: Here in this question, given a sequence of arithmetic progression (AP), we have to find the form of the progression of \[a\], \[b\], \[c\]. For this we have to simplify the given AP up to the simplest form by using arithmetic operations like addition, subtraction, multiplication and division. Then identify the resultant sequence by its nature or condition.

Complete answer:
An arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.
The fixed number that must be added to any term of an AP to get the next term is known as the common difference of the AP it will be same for the whole sequence.
Consider the given sequence AP
\[\left( {\dfrac{{b + c - a}}{a}} \right)\], \[\left( {\dfrac{{c + a - b}}{b}} \right)\], \[\left( {\dfrac{{a + b - c}}{c}} \right)\]
Add 2 to all the terms of AP, then
\[ \Rightarrow \,\,\left( {\dfrac{{b + c - a}}{a} + 2} \right)\], \[\left( {\dfrac{{c + a - b}}{b} + 2} \right)\], \[\left( {\dfrac{{a + b - c}}{c} + 2} \right)\]
Take LCM, a in 1st term, b in 2nd term and c in 3rd term, then we have
 \[ \Rightarrow \,\,\left( {\dfrac{{b + c - a + 2a}}{a}} \right)\], \[\left( {\dfrac{{c + a - b + 2b}}{b}} \right)\], \[\left( {\dfrac{{a + b - c + 2c}}{c}} \right)\]
On simplification, we get
\[ \Rightarrow \,\,\left( {\dfrac{{b + c + a}}{a}} \right)\], \[\left( {\dfrac{{c + a + b}}{b}} \right)\], \[\left( {\dfrac{{a + b + c}}{c}} \right)\]
Divide each term by \[\left( {a + b + c} \right)\], then
\[ \Rightarrow \,\,\left( {\dfrac{{b + c + a}}{a}} \right)\left( {\dfrac{1}{{a + b + c}}} \right)\], \[\left( {\dfrac{{c + a + b}}{b}} \right)\left( {\dfrac{1}{{a + b + c}}} \right)\], \[\left( {\dfrac{{a + b + c}}{c}} \right)\left( {\dfrac{1}{{a + b + c}}} \right)\]
On simplification, we get
\[ \Rightarrow \,\,\dfrac{1}{a}\], \[\dfrac{1}{b}\], \[\dfrac{1}{c}\]
We know that,
A Harmonic Progression (HP) is defined as a sequence of real numbers which is determined by taking the reciprocals of the arithmetic progression (AP) that does not contain 0.
\[\therefore \,\,\,\,\dfrac{1}{a}\], \[\dfrac{1}{b}\], \[\dfrac{1}{c}\] are in H.P.
Therefore, option C is the correct answer.

Note:
For these types of questions, students must know the definition, conditions or nature of different sequences and series like Arithmetic progression (A.P), Geometric progression (G.P) and Harmonic Progression (H.P). Remember the Harmonic progression is a reciprocal of arithmetic progression.