Question

Please solve this question and kindly explain the concept of harmonic series.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that, two numbers are

$\rm :\longmapsto\:\dfrac{a}{1 - ab} \: and \: \dfrac{a}{1 + ab}$

Now, we have to find Harmonic mean between the the given numbers.

Let assume that H be the Harmonic mean between them.

So,

$\red{\rm :\longmapsto\:\dfrac{a}{1 - ab}, \:H, \: \dfrac{a}{1 + ab} \: are \: in \: HP}$

$\pink{\rm :\longmapsto\:\dfrac{1 - ab}{a}, \:\dfrac{1}{H} , \: \dfrac{1 + ab}{a} \: are \: in \: AP}$

Since, numbers are in AP, so common difference between their consecutive terms is same.

$\rm :\longmapsto\:\dfrac{1}{H} - \dfrac{1 - ab}{a} = \dfrac{1}{H} - \dfrac{1 + ab}{a}$

$\rm :\longmapsto\:\dfrac{2}{H} = \dfrac{1 - ab}{a} + \dfrac{1 + ab}{a}$

$\rm :\longmapsto\:\dfrac{2}{H} = \dfrac{1 - ab + 1 + ab}{a}$

$\rm :\longmapsto\:\dfrac{2}{H} = \dfrac{2}{a}$

$\rm :\longmapsto\:\dfrac{1}{H} = \dfrac{1}{a}$

$\purple{\rm \implies\:\boxed{ \tt{ \: \: H \: = \: a \: \: }}}$

• Hence, option (a) is correct.

Harmonic Series :- Harmonic series or progression is defined as the sequence of real numbers whom Reciprocal are in Arithmetic progression. It is denoted shortly as HP and its general series is given by

$\rm :\longmapsto\:\dfrac{1}{a}, \: \dfrac{1}{a + d}, \: \dfrac{1}{a + 2d}, \: \dfrac{1}{a + 3d}, \: - -$

So, that,

$\rm :\longmapsto\:a, \: a + d, \: a + 2d, \: a + 3d, \: - - - \: are \: in \: AP$

and

↝ nᵗʰ term of an harmonic progression is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:= \frac{1}{\:a\:+\:(n\:-\:1)\:d} }}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

aₙ is the nᵗʰ term.

a is the first term of the arithmetic sequence.

n is the no. of terms.

d is the common difference of arithmetic sequence.

Answer 2

Answer:

Option a is correct.

Step-by-step explanation:

Using HM = 2ab/a+b