Question

If \[b + c,c + a,a + b\]are in H.P., then \[\frac{a}{{b + c}},\frac{b}{{c + a}},\frac{c}{{a + b}}\]are in
A. A.P.
B. G.P.
C. H.P.
D. None of these

Answer 1

Hint
A progression is an arrangement of numbers in mathematics that follows a predetermined pattern. It is a kind of number set that adheres to clear, unambiguous norms. The two concepts of progression and sequence are distinct from one another. The nth term of a progression is computed using a specific formula, but the nth term of a sequence is determined by a set of logical rules. Arithmetic, geometric, and harmonic progressions are the three main categories into which a progression can be categorised.
Any phrase in a harmonic progression is regarded as the harmonic mean of its two neighbours. We need to identify the corresponding arithmetic progression sum in order to solve the harmonic progression difficulties.
Formula use:
The arithmetic progression is \[a,b,c\]
\[(b - a) = (c - b)\]
If \[a,b,c\] are in HP
\[\frac{1}{a} ,\frac{1}{b} ,\frac{1}{c}\] are in A.P
Complete step-by-step solution
The given series is \[b + c,c + a,a + b\]
To find the series is \[\frac{a}{{b + c}},\frac{b}{{c + a}},\frac{c}{{a + b}}\]
The series in A.P is calculated as
\[1 + [(b + c)/a],1 + [(c + a)/b],1 + [(a + b)/c]\]
This equation can be simplified and written as
\[(a + b + c)/a,(a + b + c)/b,(a + b + c)/c\]
From this equation, it is proved that these series lies on the A.P
Then, \[a,b,c\] are determined to be in H.P.
Therefore, the correct option is C.
Note
The difference between any two consecutive integers in an arithmetic progression (AP) sequence of numbers is always the same amount. It also goes by the name Arithmetic Sequence. It is feasible to find a formula for the nth term for a specific kind of sequence called a progression. The most popular mathematical progression, with simple formulas, is the arithmetic progression. The arithmetic progression property is represented by all three terms.