Question

if the first two terms of the harmonic progression be 1 /2 and 1 / 4 the the harmonic mean of the first four numbers ​

Answer 1

Answer:-

Given :

First two terms of a HP are 1/2 ; 1/4.....

We know that,

Harmonic progression is the reciprocal of Arithmetic Progression.

Hence, 2 , 4 ... are in AP.

Here, a(1) = 2 ; a(2) = 4

→ d ( Common difference) = 4 - 2 = 2.

As we know,

nth term of an AP = a + (n - 1)d

→ a(3) = a + (3 - 1)d

→ a(3) = 2 + 2(2)

→ a(3) = 6

Similarly,

→ a(4) = 2 + 3(2)

→ a(4) = 8

Now,

Arithmetic Mean of first n terms = (Sum of n terms)/number of terms.

→ A. M = [a(1) + a(2) + a(3) + a(4)]/4

→ H.M = 4/(2 + 4 + 6 + 8) [ Reciprocal of A.M]

→ H.M = 4/20

→ H.M = 1/5

Hence, the Harmonic mean of first 4 terms is 1/5.

Answer 2

Concept Introduction: Harmonic Progression is the reverse of Arithmetic progression.

Given:

We have been Given: The first two terms of the Harmonic Progression are:

$\frac{1}{2} \\ \frac{1}{4}$

To Find:

We have to Find: Find ths Harmonic mean of First Four numbers.

Solution:

According to the problem, You will find that the denominators of Harmonic Progression is in AP. therefore the common difference between the series is

$d = 4 - 2 = 2$

Now, finding the Nth term of AP is

$a_{n} = a + (n - 1)d$

therefore the third number in the series is

$a_{3} = 2 + (3 - 1) \times 2 \\ = 2 + 2 \times 2 = 2 + 4 = 6$

therefore the fourth number in the series is,

$a_{4} = 2 + (4 - 1) \times 2 \\ = 2 + 3 \times 2 = 2 + 6 =8$

therefore the Harmonic Progression Mean will be

$hp_{mean} = \frac{4}{20} = \frac{1}{5}$

Final Answer: The Harmonic Progression Mean of first four terms is

$\frac{1}{5}$

#SPJ2