Question

If a, b, c are in A.P. then $\dfrac{a}{{bc}},\,\dfrac{1}{c},\,\dfrac{1}{b}$ are in
A) A.P.
B) G.P.
C) H.P.
D) None of these

Answer 1

In order to solve this question, you have to know about A.P. In this question a, b, c are in A.P. which means the double of the second term is the addition of the first term and third term i.e. $2b = a + c$.
Now, just convert our given equation $\dfrac{a}{{bc}},\,\dfrac{1}{c},\,\dfrac{1}{b}$ like the general equation of A.P. and you will get the correct answer.

Complete step by step answer:
We are given that the variables a, b, c are in arithmetic progression.
We know that when variables are in arithmetic progression then this means the second term is the addition of the first term and third term.
Here the first term is $a$, the second term is $b$ and the third term is $c$.
Which means $2b = a + c$ .
And we have found $\dfrac{a}{{bc}},\,\dfrac{1}{c},\,\dfrac{1}{b}$ are in which progress.
$\Rightarrow 2b = a + c$
Now, we have to divide by $bc$ on both the side of above equation and we will find our new equation looks similar to general equation of arithmetic progression,
Divide by $bc$
$\Rightarrow \dfrac{2}{c} = \dfrac{a}{{bc}} + \dfrac{1}{b}$
Now, take least common multiple (L.C.M.) of above equation and we will get,
$\Rightarrow \dfrac{2}{c} = \dfrac{{a + c}}{{bc}}$
Let’s cancel $c$ from both the side and we will get,
$\Rightarrow 2 = \dfrac{{a + c}}{b}$
Take $b$ to the left side and finally we will see beloved equation,
$\Rightarrow 2b = a + c$
See this is the equation of arithmetic progression.
Therefore, we can clearly see that $\dfrac{a}{{bc}},\,\dfrac{1}{c},\,\dfrac{1}{b}$ are in A.P.
So, the correct option is (A).

Note:
Relation between arithmetic progression (A.P.), geometric progression (G.P.) and harmonic progression (H.P.): If A.M denotes the arithmetic mean, G.M denotes the geometric mean and H.M, the harmonic mean, then the relationship between the three is given by,
$A.M. \times G.M. = {(H.M.)^2}$