Harmonic Progression (HP) is an important concept in algebra and sequences, especially in competitive exams and real-life problems involving rates, speed, and work. Unlike AP or GP, an HP does not follow a direct pattern in its original form. Instead, the sequence becomes meaningful when we take the reciprocals of its terms. Once converted, those reciprocals form an Arithmetic Progression (AP), making it easy to calculate any term using AP formulas. Understanding HP makes solving rate-based questions faster and more accurate.
Formula for Calculating Harmonic Progression (HP) – Overview
| Formula | Variables | When it is Used |
|---|---|---|
| Tn = 1 / [a + (n – 1)d] | a = first term of AP, d = common difference, n = number of terms | Used to find any term in a Harmonic Progression |
What is Harmonic Progression in Maths?
A Harmonic Progression (HP) is a sequence where the reciprocals of the terms form an Arithmetic Progression (AP). HP is not directly defined like AP or GP; instead, it is derived by converting the given terms into their reciprocals.
Example:
The sequence 1/2, 1/4, 1/6, 1/8 is an HP because their reciprocals 2, 4, 6, 8 form an AP.
To find terms in HP:
- Convert HP terms into reciprocals.
- These reciprocals form an AP with first term a and common difference d.
- Use the AP nth term formula: a + (n – 1)d.
- Take the reciprocal again to get the HP term.
HP is extremely useful in rate-based calculations, such as average speed, time-and-work, and mathematical aptitude questions asked in CUET, JEE, SSC, Banking, and other competitive exams.
Examples to Calculate Harmonic Progression (HP)
Example 1: Find the 5th term of the HP whose corresponding AP is 2, 4, 6, …
Step 1: For the AP: a = 2, d = 2
Step 2: nth term of AP = a + (n – 1)d
= 2 + (5 – 1) × 2
= 2 + 8 = 10
Step 3: nth term of HP = 1 / 10
Answer: The 5th term of the HP is 1/10.
Example 2: Find the 4th term of the HP whose corresponding AP is 1, 3, 5, …
Step 1: For the AP: a = 1, d = 2
Step 2: nth term of AP = a + (n – 1)d
= 1 + (4 – 1) × 2
= 1 + 6 = 7
Step 3: nth term of HP = 1 / 7
Answer: The 4th term of the HP is 1/7.
FAQs about Harmonic Progression Formula
Q1. How do you identify if a sequence is in HP?
If the reciprocals of its terms form an AP, then the sequence is an HP.
Q2. Is there a direct formula for the sum of an HP?
No, HP does not have a direct sum formula. We first convert it to AP to calculate sums.
Q3. Where do we use HP in real life?
HP is commonly used in average speed questions, work-time problems, and rate-based calculations.
Q4. What is the difference between AP, GP, and HP?
AP has a constant difference, GP has a constant ratio, and HP is formed when the reciprocals of the terms form an AP.
Q5. Is HP important for exams?
Yes, questions on HP appear frequently in CUET, SSC, Banking, JEE, and other aptitude tests.