# Harmonic Progression Shortcut, tricks, and tips

## Harmonic Progression tricks, shortcuts, and tips

In this page tips and tricks for Harmonic Progression are given along with different types of Questions.

Here, are some easy tips and tricks for you on HP with easy, and efficiently used tricks in competitive and recruitment exams.

### Harmonic Progression Tips and Tricks and Shortcuts

• For two numbers, if A, G and H are respectively the arithmetic, geometric and harmonic means, then
A≥ G ≥ H
• Relationship between arithmetic, geometric, and harmonic means $AM \times HM = GM^{2}$
• Unless a = 1 and n = 1, the sum of a harmonic series will never be an integer. This is because at least one denominator of the progression is divisible by a prime number that does not divide any other denominator.
• Three consecutive numbers of a HP are: $\mathbf{\frac{1}{(a–d)}, \frac{1}{a}, \frac{1}{(a+d)}}$
• Four consecutive numbers of a HP are: $\mathbf{\frac{1}{(a–3d)}, \frac{1}{(a–d)}, \frac{1}{(a+d)}, \frac{1}{(a+3d)}}.$

### Type 1: Find nth term of series $a_{n} = \frac{1}{a+(n-1)d}$

Question 1 Find the 8th term in the series $\frac{1}{2}, \frac{1}{4}, \frac{1}{6}$…….

Options:

A. $\frac{1}{20}$

B. $\frac{1}{14}$

C. $\frac{1}{16}$

D. $\frac{1}{18}$

Solution:    We know that,
${a_{n}} = \frac{1}{a+(n-1)d}$

Convert the HP series in AP

We get 2, 4, 6,……

In the given series,

a (first term) = 2

d (common difference) = 2 …. (4 – 2)

Therefore, 8th term = t8 = a + (n-1) d

t8 = 2 + (8– 1) 2

t8 = 2 + 7 x 2

t8 = 2 + 14

t8 = 16

Correct option: C

Question 2  Find the Middle term in this Harmonic series 4 , a , 6

Options:

A. $\frac{22}{5}$

B. $\frac{21}{5}$

C. $\frac{23}{5}$

D. $\frac{24}{5}$

Solution:    We know that,
${a_{n}} = \frac{1}{a+(n-1)d}$

we know Inverse of HP is AP.

So, $\frac{1}{4} , \frac{1}{a} ,\frac{1}{6}$ is in AP

$\frac{1}{a} – \frac{1}{4} = \frac{1}{6} – \frac{1}{a}$

a = $\frac{24}{5}$

Correct option: D

### 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

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