Question

If \[a,b,c\] 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
Before solving the problem, we have to know about A.P. and H.P.
Taking the reciprocals of the arithmetic progression that does not contain zero yields a sequence of real numbers known as a harmonic progression (HP). Any phrase in a harmonic progression is regarded as the harmonic mean of its two neighbors. An arithmetic progression is a set of integers with a fixed difference between any two succeeding numbers (A.P.).
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.
Formula used:
If a,b,c are in HP then
\[\frac{1}{a} + \frac{1}{c} = \frac{2}{b}\]

Complete step-by-step solution
The coordinates are \[a,b,c\]
Then these points are in harmonic progression (H.P.)
\[\frac{1}{a} + \frac{1}{c} = \frac{2}{b}\]
This Points are also present in Arithmetic progression (A.P)
\[ = > \frac{{a + b + c}}{a},\frac{{a + b + c}}{b},\frac{{a + b + c}}{c}\]
This then is solved as the points in A.P
\[ = > \frac{{b + c}}{a},\frac{{a + c}}{b},\frac{{a + b}}{c}\]
And for the points in H.P, the points are reciprocated as
\[ = > \frac{a}{{b + c}},\frac{b}{{a + c}},\frac{c}{{a + b}}\]
Therefore, the correct option is C.
Note:
A series of terms is referred to as a geometric progression if each next term is produced by multiplying each previous term by a fixed amount. (GP), whereas the common ratio is the name given to the constant value. Since the Sum of GP is smaller than the Sum of AP even in real numbers, the Sum of GP cannot be more than the Sum of AP in a series where the domain is integers. When the reciprocals of the elements are in arithmetic progression, a group of words is referred to as an HP series.