Understanding Harmonic Progression (HP)

Understanding Harmonic Progression (HP)

A Harmonic Progression (HP) is a sequence formed by taking the reciprocals of an Arithmetic Progression (AP). In other words, the terms of an HP sequence are the reciprocals of the terms of an AP sequence.

Formula of Harmonic Progression

If a,a+d,a+2d,…a, a+d, a+2d, \dots is an arithmetic progression, then the corresponding harmonic progression is:1a,1a+d,1a+2d,…\frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}, \dotsWhere:
  • aa is the first term of the AP.
  • dd is the common difference of the AP.

Harmonic Mean: The Core of HP

The Harmonic Mean is a key concept when dealing with HP. It is calculated using the following formula:H=2aba+bH = \frac{2ab}{a + b}This is particularly useful for finding the average of rates, such as average speed, when the values of aa and bb represent rates.

Extended Formula for Harmonic Mean of Multiple Numbers:

For more than two numbers, the harmonic mean is:H=n(1a1+1a2+⋯+1an)H = \frac{n}{\left(\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}\right)}Where:
  • a1,a2,…,ana_1, a_2, \dots, a_n are the numbers involved.

Applications of Harmonic Mean in Real Life

Average Speed

If a vehicle covers the same distance at two different speeds, xx and yy, the average speed is the harmonic mean of xx and yy. For example, if the speeds are 40 km/h and 60 km/h, the average speed is 48 km/h.

In Triangles

  • In any triangle, the radius of the incircle is one-third of the harmonic mean of the altitudes.
  • In right-angled triangles, the altitude from the right angle is half the harmonic mean of the squares of the legs.

Solved Examples for Harmonic Progression

Example 1: Find the 6th Term in an HP

Problem: The sum of the reciprocals of the first 11 terms of an HP series is 110. Find the 6th term.Solution: The reciprocals of the first 11 terms of an HP make an AP. We can use the formula for the sum of an AP:Sn=n2[2a+(n−1)d]S_n = \frac{n}{2} [2a + (n-1)d]Given that S11=110S_{11} = 110, we calculate a+5d=10a + 5d = 10, which is the 6th term of the corresponding AP. Therefore, the 6th term in HP is:6th term of HP=110\text{6th term of HP} = \frac{1}{10}

Example 2: Find the 16th Term of an HP

Problem: The 6th and 11th terms of an HP are 10 and 18, respectively. Find the 16th term.Solution: We first express the terms of HP in terms of AP and solve for aa and dd:6th term of AP:a+5d=110\text{6th term of AP:} a + 5d = \frac{1}{10} 11th term of AP:a+10d=118\text{11th term of AP:} a + 10d = \frac{1}{18}By solving these, we find a=1390a = \frac{13}{90} and d=−2225d = \frac{-2}{225}.The 16th term is:16th term=11390−2225=90\text{16th term} = \frac{1}{\frac{13}{90} – \frac{2}{225}} = 90

Trick to Solve HP Problems Faster

To make HP problems simpler, you can use the following trick:
  1. Express the given HP terms as an AP by taking reciprocals.
  2. Solve using standard AP formulas.

FAQs on Harmonic Progression

How to Find the nth Term of a Harmonic Progression?

The nth term of a harmonic progression is given by:Tn=1a+(n−1)dT_n = \frac{1}{a + (n-1)d}Where aa is the first term, dd is the common difference of the corresponding AP.

How Do You Solve HP Problems Involving Distinct Numbers?

If three distinct numbers a,b,ca, b, c are in HP, the relationship between them can be expressed as:1a,1b,1c are in AP.\frac{1}{a}, \frac{1}{b}, \frac{1}{c} \text{ are in AP.}By substituting the terms into the AP formula, we can solve for the desired values.

Visual Suggestions for the Article:

  1. Visual 1: Harmonic Progression and Arithmetic Progression Relationship A diagram showing an HP sequence derived from an AP sequence.
  2. Visual 2: Harmonic Mean Formula A simple graphic of the harmonic mean formula with a practical example (e.g., average speed).
  3. Visual 3: Solving HP Problems A flowchart of how to solve typical HP problems using the AP method.

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