Question

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

Answer 1

Hint
The definition of reciprocal in mathematics is the inverse of a value or a number. Reciprocals are used to convert division to multiplication when dividing fractions, which simplifies the division process. In mathematics, a set of numbers is referred to as an HP if the reciprocals of the terms are in AP. The AP cost per invoice is calculated by dividing the total number of invoices paid during a specified period of time by the total costs incurred to pay those invoices.
This indicator offers a precise assessment of a company's AP efficiency, together with other accounts payable KPIs. Graphical Progress (GP) Geometric progression is a sequence where the ratio of two consecutive terms is constant.
Formula use:
If \[a,b,c\] are in AP
\[(b - a) = (c - b)\]
If \[a,b,c\] are in GP
\[{b^2} = ac\].
If \[a,b,c\] are in HP
\[\frac{1}{a} + \frac{1}{c} = \frac{2}{b}\]
Complete step-by-step solution
The given equation is \[\frac{a}{b},\frac{b}{c},\frac{c}{a}\]
Which is in the H.P progression series.
Then, the reciprocal of this equation will be in A.P.
That is, \[\frac{b}{a},\frac{c}{b},\frac{a}{c}\]
The common difference between the equation are noted as
\[\frac{c}{b} - \frac{b}{a} = \frac{a}{c} - \frac{c}{b}\]
This then becomes,
\[ = > \frac{{(ac - {b^2})}}{{ab}} = \frac{{(ab - {c^2})}}{{bc}}\]
The value b will be eliminated from both the sides
\[c(ac - {b^2}) = a(ab - {c^2})\]
\[ = > {c^2}a - {b^2}c = {a^2}b - {c^2}a\]
The equation is solved as
\[ = > 2{c^2}a = {a^2}b + {b^2}c\]
So, the progression series are calculated to be in A.P.
Therefore, the correct option is A.

Note
A harmonic progression is a set of numbers that contains the reciprocals of the terms in an arithmetic progression. The definition of reciprocal in mathematics is the inverse of a value or a number. The reciprocal of the nth term in the equivalent arithmetic progression is the nth term in a harmonic progression.
Each term in an arithmetic-geometric progression (AGP) can be modelled as the union of the terms from an arithmetic progression (AP) and a geometric progression (GP) (GP).