Question

If there are four harmonic means between 1/12, 1/42,
then the third harmonic mean is​

Answer 1

If there are four harmonic means between 1/12, 1/42,then the third harmonic mean is 1/30

Step-by-step explanation:

Let the 4 harmonic means are

$H_1, H_2, H_3, H_4$

Then

$1/12, H_1, H_2, H_3, H_4, 1/42$ are in HP

or, $\frac{12}{1}, \frac{1}{H_1}, \frac{1}{H_2}, \frac{1}{H_3}, \frac{1}{H_4}, 42$ are in AP

Let $\frac{1}{H_1}=A_1,\frac{1}{H_2}=A_2,\frac{1}{H_3}=A_3,\frac{1}{H_4}=A_4$

Thus,

$12, A_1, A_2, A_3, A_4, 42$ are in AP

First term of this AP ,  a = 12

42 will be the 6th term of the AP

Let the common difference be d

Then nth term of AP is given by

$A_n=a+(n-1)d$

$A_6=12+(6-1)\times d$

$\implies 42=12+5d$

$\implies 5d=30$

$\implies d=6$

Therefore,

$A_3=12+(4-1)\times 6$

$\implies A_3=12+18$

$\implies A_3=30$

Thus, $H_3=\frac{1}{A_3}$

or. $H_3=\frac{1}{30}$

Therefore, the third harmonic mean is 1/30

Hope this answer is helpful.

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