Harmonic Progression Shortcut, tricks, and tips

Type 2: Find the Harmonic mean of the series.\mathbf{\frac{n} {\frac{1}{a_{1}} + \frac{1}{a_{2}} ……\frac{1}{a_{n}}}}

Question 1. Find the harmonic mean (HM) of 8, 9, 6?
Options:

A. 4.55

B.6.65

C. 7.56

D. 7.44

Solution:    We know that,

H.M = \frac{3abc}{ab + bc + ca}

HM = \frac{3\times 8 \times 9 \times 6}{8\times 9 + 9\times 6 +6\times 8 }

HM = 7.44

Correct option: D

Question 2. If Harmonic Mean of two numbers is 8 and one of the number is 12 then Find Another Number.

Options:

A. 2
B. 4
C. 6
D. 8

Solution:   

We know that,
H.M = \frac{2ab}{a+b}
Let another number = b

8 = \frac{2\times 12\times b}{12+b}

96 + 8b = 24b

b = \frac{96}{16} = 6

Another number = 6

Correct option: C

Question 3. Find the correct option for A, G , H for the elements 4 and 6 . where A is Arithmetic Mean ,G is Geometric Mean & H is Harmonic Mean.

Options:

 A. A ≥ G ≥ H

B. A < G > H

C. A < G ≥ H

D. A < G < H

Solution      Arithmetic mean (A) = \frac{4 +6}{2} = 5

Geometric mean (G) = (4\times 6)^{\frac{1}{2}} = 4.89

Harmonic mean (H) = \frac{2}{( \frac{1}{4} + \frac{1}{6})} = 4.8

Hence it is shown that A ≥ G ≥ H

Correct option: A