Question

Which of the following is in the form of H. P.?
A.1, 2, 4, 8, 16, ....
B.2, 4, 6, 8, 10, ….
C.$\dfrac{1}{3}$, $\dfrac{2}{3}$, 1, $\dfrac{4}{3}$, ….
D.- 1, 1, - 1, 1, - 1, ….

Answer 1

Hint: We will use the definition of the harmonic progressions H. P. to determine if the given options form an H. P. or not. We will check if the reciprocals of the given H.P. forms an A. P. or not. If they form, then they will be considered to form a H. P. If not, then we will discard the option.

Complete step-by-step answer:
We are given options and we need to check if they form harmonic progression H. P.
Definition of H. P.: In mathematics, a harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression.
Or in other simpler words, a series of terms is known as a H. P. if their reciprocals are in an arithmetic progression A. P.
Option(A): 1, 2, 4, 8, 16, ....
Reciprocal of this series will be $1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},\dfrac{1}{{16}},....$
If they form an A. P., they must have a common difference d.
$ \Rightarrow $$\dfrac{1}{2} - 1 = - \dfrac{1}{2}$, $\dfrac{1}{4} - \dfrac{1}{2} = - \dfrac{1}{4}$, $\dfrac{1}{8} - \dfrac{1}{4} = - \dfrac{1}{8}$, ..
As we can see that the difference is not equal, hence this is not an A.P. and therefore, option(A) is not a H.P.
Option(B): 2, 4, 6, 8, 10, ….
Reciprocal will be $\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{6},\dfrac{1}{8},\dfrac{1}{{10}},....$
Their difference is: $\dfrac{1}{4} - \dfrac{1}{2} = - \dfrac{1}{4}$, $\dfrac{1}{6} - \dfrac{1}{4} = - \dfrac{1}{{12}}$,…
Hence, we can say that this is not an A.P. and therefore, option(B) is not a H.P.
Option(C): $\dfrac{1}{3}$, $\dfrac{2}{3}$, 1, $\dfrac{4}{3}$, ….
Their reciprocal will be: $3,\dfrac{3}{2},1,\dfrac{3}{4},....$
Their difference will be:$\dfrac{3}{2} - 3 = - \dfrac{3}{2}$, $1 - \dfrac{3}{2} = - \dfrac{1}{2}$,…
Hence, option(C) cannot be in a H.P. because the reciprocal of its terms is not in an A.P.
Option(D): - 1, 1, - 1, 1, - 1, ….
Their reciprocals will be: - 1, 1, - 1, 1, - 1, ….
Their difference will be: $1 - \left( { - 1} \right) = 2$, $ - 1 - 1 = - 2$
Hence, it is also not in an A.P.
Therefore, option (D) is also not in an H.P.
Hence none of the options are correct.

Note: We can see that in this problem, you may get confused in how to check if they are in H.P. or not. You must know the definition because such questions are generally easy and definition based. You can also solve this question by using the formula of the nth term of a H.P.