# Tips And Tricks And Shortcuts For Harmonic Progression

## Tips and Tricks for Harmonic Progression

Harmonic Progression is a series of values or numbers, when their reciprocals are in Arithmetic Progression. Harmonic Progression is also known as Harmonic Sequence. Here are some of the best Tips and Tricks and Shortcuts to solve Harmonic progression more effectively.

There are Many Applications of Harmonic Progression. It is used in Geometry and Time , Speed and Distance also. In this Page Tips and Tricks for Harmonic Progression along with different types of Questions is given.

### Tips and Tricks And Shortcuts on  harmonic progression (HP)

• Here, are Some easy Tips and Tricks for you on HP with easily, and efficiently used tricks in Competitive and Recruitment exams.
• There are 2 types of questions asked in exams.

### 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