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Harmonic Progression (H.P.)

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Introduction

Harmonic Progression (H.P.) वह sequence है जिसमें दिए गए terms के reciprocals एक Arithmetic Progression (A.P.) बनाते हैं। H.P. वाले questions अक्सर ratio, speed-time और mixture जैसे topics में आते हैं। Reciprocals को A.P. में बदलकर pattern समझना बहुत आसान हो जाता है।

Pattern: Harmonic Progression (H.P.)

Pattern: Harmonic Progression (H.P.)

Key idea: अगर किसी H.P. के reciprocals एक A.P. बनाते हैं जिसका first term A और difference D है, तो H.P. का nth term होगा:

Tₙ = 1 / (A + (n - 1)D)

Practically हम दो तरीकों से solve करते हैं: (a) H.P. → reciprocals → A.P. solve करें, या (b) अगर A.P. साफ दिख रहा हो तो सीधे formula use करें।

Step-by-Step Example

Question

Sequence 1/3, 1/5, 1/7, 1/9, … एक H.P. है। इसका 5वाँ term निकालें।

Solution

  1. Step 1: Reciprocals लेकर underlying A.P. पहचानें

    Reciprocals: 3, 5, 7, 9, … → यह एक A.P. है जिसमें first term A = 3 और common difference D = 2 है।

  2. Step 2: A.P. का nth-term formula लगाएँ

    A.P. का nth term = A + (n - 1)D n = 5 के लिए: A₅ = 3 + (5 - 1)×2 = 3 + 8 = 11

  3. Step 3: वापस H.P. term पाने के लिए reciprocal लें

    T₅ = 1 / 11

  4. Final Answer:

    5वाँ term = 1/11

  5. Quick Check:

    List: 1/3, 1/5, 1/7, 1/9, 1/11 → पाँचवाँ term सही है। ✅

Quick Variations

1. यदि H.P. terms 1/(a + kd) के रूप में हों, तो सीधे a और d पहचानकर nth term निकालें।

2. दो H.P. terms (जैसे Tₚ और T_q) दिए हों → reciprocals लेकर A.P. parameters निकालें और फिर वापस convert करें।

3. Story/problem में missing H.P. term निकालने के लिए reciprocals को A.P. में बदलें।

Trick to Always Use

  • Step 1 → हमेशा पहले reciprocals लें: इससे pattern साफ दिखता है।
  • Step 2 → फिर A.P. के formulas (nth term, sum आदि) लागू करें और बाद में reciprocal लेकर H.P. term पाएं।

Summary

Harmonic Progression के मुख्य बिंदु:

  • H.P. की पहचान: उसके reciprocals एक A.P. बनाते हैं।
  • General nth term: Tₙ = 1 / (A + (n - 1)D).
  • Word problems में rates/ratios होने पर values को reciprocals में बदलना आसान solution देता है।
  • हमेशा reciprocal लेकर term को वापस confirm करें।

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.