Tips and Tricks: Harmonic Progressions

# Tips and Tricks: Harmonic Progressions | Quantitative Aptitude for SSC CGL PDF Download

## HP tricks

One useful technique is to convert the HP into an AP of reciprocals. This can simplify calculations, as working with an AP might be more familiar and straightforward.

## 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 × HM = GM2
•  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:
• Four consecutive numbers of a HP are:

### Type 1: Find nth term of series

Example 1: Find the Middle term in this Harmonic series 4, a, 6
(a) 22/5
(b) 21/5
(c) 23/5
(d) 24/5
Ans:
(d)
We know that,

we know Inverse of HP is AP.

Example 2: Find the 8th term in the series 1/2, 1/4, 1/6  …….
(a) 1/20
(b) 1/14
(c) 1/16
(d) 1/18
Ans:
(c)
We know that,

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

### Type 2: Find the Harmonic mean of the series

Example 1: 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.
(a) A ≥ G ≥ H
(b) A < G > H
(c) A < G ≥ H
(d) A < G < H
Ans:
(a)
Arithmetic mean
Geometric mean (G) = ( 4 × 6 )1/2 = 4.89

Hence it is shown that A ≥ G ≥ H

Example 2: If Harmonic Mean of two numbers is 8 and one of the number is 12 then Find Another Number.
(a) 2
(b) 4
(c) 6
(d) 8
Ans:
(c)

We know that,

Let another number = b

96 + 8b = 24b

Another number = 6

Example 3: Find the harmonic mean (HM) of 8, 9, 6?
(a) 4.55
(b) 6.65
(c) 7.56
(d) 7.44
Ans:
(d)
We know that,

HM = 7.44

The document Tips and Tricks: Harmonic Progressions | Quantitative Aptitude for SSC CGL is a part of the SSC CGL Course Quantitative Aptitude for SSC CGL.
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## Quantitative Aptitude for SSC CGL

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## FAQs on Tips and Tricks: Harmonic Progressions - Quantitative Aptitude for SSC CGL

 1. What is a harmonic progression?
A harmonic progression is a sequence of numbers in which the reciprocal of each term is in arithmetic progression. In other words, the ratio between consecutive terms remains constant.
 2. How can harmonic progressions be useful in mathematics?
Harmonic progressions are widely used in various branches of mathematics, including calculus, number theory, and music theory. They help in understanding the behavior of functions, series, and sequences, as well as in solving problems related to ratios and proportions.
 3. What are some common examples of harmonic progressions?
Some common examples of harmonic progressions include the sequence 1, 1/2, 1/3, 1/4, 1/5, and so on. Another example is the series 1, 1/3, 1/5, 1/7, 1/9, which consists of the reciprocals of odd numbers.
 4. How can one determine the sum of a harmonic progression?
The sum of a harmonic progression can be found using the formula: S = n/1 + n/2 + n/3 + ... + n/n, where n is the number of terms in the progression. This sum is often denoted as Hn, and its value can be approximated using the natural logarithm function.
 5. Are there any shortcuts or tricks for working with harmonic progressions?
Yes, there are some shortcuts and tricks that can be applied when dealing with harmonic progressions. For example, if the terms of the progression are reciprocals of consecutive integers, the sum can be simplified to n/2. Additionally, if the terms are reciprocals of consecutive odd numbers, the sum can be simplified to n/4. These shortcuts can save time and effort in calculations.

## Quantitative Aptitude for SSC CGL

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