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Harmonic Progression (H.P.)

Introduction

A Harmonic Progression (H.P.) is a sequence where the reciprocals of its terms form an Arithmetic Progression (A.P.). H.P. problems often appear in ratio, speed-time, and mixture questions - recognizing the A.P. behind the reciprocals makes them easy to handle.

Pattern: Harmonic Progression (H.P.)

Pattern

Key idea: If the reciprocals of terms form an A.P. with first term A and difference D, then the nth term of the H.P. is

Tₙ = 1 / (A + (n - 1)D)

In practice you either: (a) convert H.P. → reciprocals → solve as A.P., or (b) use the above formula when the underlying A.P. is visible.

Step-by-Step Example

Question

The sequence 1/3, 1/5, 1/7, 1/9, … is a Harmonic Progression. Find the 5th term.

Solution

  1. Step 1: Identify the underlying A.P. (reciprocals)

    Take reciprocals of the H.P. terms: 3, 5, 7, 9, … . These form an A.P. with first term A = 3 and common difference D = 2.

  2. Step 2: Use the nth-term formula for the underlying A.P.

    The nth term of the underlying A.P. is A + (n - 1)D. For n = 5: A₅ = 3 + (5 - 1)×2 = 3 + 8 = 11.

  3. Step 3: Convert back to H.P. term by taking reciprocal

    The 5th term of the H.P. is the reciprocal of A₅: T₅ = 1 / 11.

  4. Final Answer:

    The 5th term is 1/11.

  5. Quick Check:

    List terms to confirm: 1/3, 1/5, 1/7, 1/9, 1/11 - yes, the 5th term is 1/11 ✅

Quick Variations

1. If H.P. is given in terms like 1/(a + kd), identify a and d directly.

2. Sometimes problems give two H.P. terms (e.g., Tₚ and T_q) - convert to reciprocals, find the A.P. parameters, then convert back.

3. If a term of H.P. is missing in a story problem, convert to A.P. of reciprocals to solve.

Trick to Always Use

  • Step 1 → Flip to reciprocals: Always check if taking reciprocals gives a clear A.P.
  • Step 2 → Use A.P. tools: Apply A.P. formulas (nth term, sum, difference) to the reciprocals, then flip back.

Summary

Summary

Key takeaways for Harmonic Progression:

  • H.P. = reciprocals form an A.P.; work with the reciprocals to simplify calculations.
  • General nth term (when reciprocals are A.P. with A and D): Tₙ = 1 / (A + (n - 1)D).
  • For word problems, translate quantities into reciprocals if rates or ratios appear.
  • Quick check: always reconvert a computed reciprocal back to the H.P. term to confirm the sequence.

Practice

(1/5)
1. Find the 4th term of the Harmonic Progression: 1/2, 1/4, 1/6, …
easy
A. 1/8
B. 1/10
C. 1/12
D. 1/14

Solution

  1. Step 1: Convert H.P. to A.P. (reciprocals)

    Reciprocals form the A.P.: 2, 4, 6, … with first term a = 2 and difference d = 2.

  2. Step 2: Find the 4th term of the A.P.

    T₄ = a + (n - 1)d = 2 + (4 - 1)×2 = 2 + 6 = 8.

  3. Step 3: Convert back to H.P.

    The 4th term of the H.P. is the reciprocal: 1/8.

  4. Final Answer:

    4th term = 1/8 → Option A.

  5. Quick Check:

    Sequence: 1/2, 1/4, 1/6, 1/8 - fourth term is 1/8 ✅

Hint: Find nth term of the reciprocal A.P., then take its reciprocal.
Common Mistakes: Applying differences directly to H.P. terms instead of to reciprocals.
2. If the reciprocals of the terms 1/3, 1/5, 1/7, … form an A.P., find the 5th term of the H.P.
easy
A. 1/9
B. 1/10
C. 1/11
D. 1/12

Solution

  1. Step 1: Identify the reciprocal A.P.

    Reciprocals → 3, 5, 7, … so a = 3 and d = 2.

  2. Step 2: Find 5th term of the A.P.

    T₅ = a + (5 - 1)d = 3 + 8 = 11.

  3. Step 3: Take reciprocal for H.P.

    H.P. 5th term = 1/11.

  4. Final Answer:

    5th term = 1/11 → Option C.

  5. Quick Check:

    H.P. sequence: 1/3,1/5,1/7,1/9,1/11 - confirms 1/11 ✅

Hint: Flip to reciprocals first to work with an A.P., then flip back.
Common Mistakes: Trying to apply A.P. formula to H.P. terms directly.
3. Find the 6th term of the Harmonic Progression: 1, 1/2, 1/3, 1/4, …
easy
A. 1/5
B. 1/6
C. 1/7
D. 1/8

Solution

  1. Step 1: Form the reciprocal A.P.

    Reciprocals: 1,2,3,4,… so a = 1, d = 1.

  2. Step 2: Compute 6th term of the A.P.

    T₆ = a + (6 - 1)d = 1 + 5 = 6.

  3. Step 3: Convert back to H.P.

    6th term of H.P. = 1/6.

  4. Final Answer:

    6th term = 1/6 → Option B.

  5. Quick Check:

    Sequence: 1,1/2,1/3,1/4,1/5,1/6 - 6th term is 1/6 ✅

Hint: H.P. nth term = reciprocal of nth term of the underlying A.P.
Common Mistakes: Confusing index when converting back and forth between H.P. and A.P.
4. If the reciprocals of an H.P. form an A.P. with first term a = 1 and difference d = 4, find the 4th term of the H.P.
medium
A. 1/10
B. 1/11
C. 1/12
D. 1/13

Solution

  1. Step 1: Find 4th term in reciprocal A.P.

    T₄ = a + (n - 1)d = 1 + (4 - 1)×4 = 1 + 12 = 13.

  2. Step 2: Take reciprocal to get H.P.

    4th term of H.P. = 1/13.

  3. Final Answer:

    4th term = 1/13 → Option D.

  4. Quick Check:

    Reciprocal sequence: 1,5,9,13 → H.P.: 1,1/5,1/9,1/13 ✅

Hint: Compute the A.P. nth term first (a + (n-1)d), then reciprocate.
Common Mistakes: Mistaking n×d for (n-1)d when using the A.P. formula.
5. If the first term of the reciprocal A.P. is 4 and the common difference is 2, find the 5th term of the corresponding H.P.
medium
A. 1/12
B. 1/10
C. 1/13
D. 1/14

Solution

  1. Step 1: Compute 5th term of the reciprocal A.P.

    T₅ = a + (5 - 1)d = 4 + 8 = 12.

  2. Step 2: Take reciprocal to get H.P.

    5th term of H.P. = 1/12.

  3. Final Answer:

    5th term = 1/12 → Option A.

  4. Quick Check:

    Underlying A.P.: 4,6,8,10,12 → H.P.: 1/4,1/6,1/8,1/10,1/12 ✅

Hint: After finding the A.P. nth term, always reciprocate to avoid sign/placement errors.
Common Mistakes: Using the wrong index for n when computing the A.P. nth term.

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