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Quantitative Aptitude · Sequence and Series

Harmonic Progression and Relation among AM, GM, HM

Updated 1 October 2026 · Fact-checked

A harmonic progression (HP) is a sequence whose reciprocals form an AP. To solve an HP question, flip every term, solve it as an AP, then flip the answer back. For two positive numbers a and b, AM = (a+b)/2, GM = √(ab), HM = 2ab/(a+b), and GM² = AM × HM.

Understand Harmonic Progression and Relation among AM, GM, HM

A harmonic progression (HP) is a sequence of non-zero numbers whose reciprocals are in AP. For example, 1, 1/2, 1/3, 1/4 is an HP because 1, 2, 3, 4 is an AP.

There is no direct formula for the sum of an HP, and you do not need one. Every HP question is an AP question in disguise. Take reciprocals, use AP rules, and take the reciprocal again at the end.

The harmonic mean (HM) of two numbers a and b is the middle term of an HP: a, H, b. Since 1/a, 1/H, 1/b are in AP, 2/H = 1/a + 1/b. This gives H = 2ab/(a+b).

For two positive numbers, the three means are linked. The arithmetic mean is (a+b)/2. The geometric mean is √(ab). The harmonic mean is 2ab/(a+b). Multiply AM by HM and you get ab, which is GM². So GM is the geometric mean of AM and HM.

For positive numbers, AM ≥ GM ≥ HM. Equality holds only when all the numbers are equal. This lets you compare values and eliminate options quickly.

Key formulas to remember

Definition of HP
a, b, c are in HP ⇔ 1/a, 1/b, 1/c are in AP
All terms must be non-zero.
nth term of HP
Tn = 1 ÷ [1/a + (n − 1)d], where a is the first term of the HP and d = 1/T2 − 1/T1
d is the common difference of the reciprocal AP, not of the HP.
HM of two numbers
H = 2ab ÷ (a + b)
Also H = 2 ÷ (1/a + 1/b).
HM of n numbers
H = n ÷ (1/a1 + 1/a2 + ... + 1/an)
Numbers must be positive for the inequality to apply.
AM, GM, HM of two numbers
AM = (a + b)/2; GM = √(ab); HM = 2ab/(a + b)
Use for positive a and b.
Relation among means
GM² = AM × HM
Holds for two positive numbers. Not for three or more in general.
Inequality
AM ≥ GM ≥ HM
For positive numbers; equal only if all numbers are equal.

How to solve Harmonic Progression and Relation among AM, GM, HM questions

Use this method for any HP or AM-GM-HM question. The idea is to convert to an AP and work with reciprocals.

  1. 1Check whether the question says HP or gives HM. If yes, plan to take reciprocals.
  2. 2Write the reciprocals of the given terms. They now form an AP.
  3. 3Find the first term and the common difference d of this AP using d = (second − first).
  4. 4Use the AP formula, such as nth term = a + (n − 1)d, on the reciprocals.
  5. 5Take the reciprocal of your result to get back to the HP term.
  6. 6For a means question, write AM, GM, HM with the formulas and use GM² = AM × HM to find the missing one.
  7. 7Check the answer against AM ≥ GM ≥ HM and against the options.

Quickest way: Flip, solve as AP, flip back; use GM² = AM × HM

When to use it: Use in the MCQ paper whenever an HP, HM, or two of the three means are given.

  1. For a missing mean, use GM² = AM × HM directly instead of computing from the numbers.
  2. If AM and GM are given, find HM = GM² ÷ AM in one step.
  3. For the nth term, find the reciprocals of two terms, get d by subtraction, and jump to the term you need.
  4. Eliminate options using AM ≥ GM ≥ HM. An HM larger than the AM is always wrong for unequal positive numbers.
  5. Try the options with small integers first: reciprocals of an HP are often whole numbers.
  6. If the reciprocal work needs more than two minutes, mark the question and move on, because wrong answers cost 0.25 marks.

Common mistakes in Harmonic Progression and Relation among AM, GM, HM

  • Applying the AP formula directly to the HP terms.

    The terms look like a sequence and students forget they must be flipped first.

    Fix: Write the reciprocals on the first line every time. Work only with them, then flip the result.

  • Forgetting to take the final reciprocal.

    The AP answer looks like a clean number, so students stop early.

    Fix: Label your answer as the term of the AP, then write 'HP term = 1 ÷ that value' before choosing an option.

  • Computing the common difference as the difference of HP terms.

    Students use d = T2 − T1 out of habit.

    Fix: Use d = 1/T2 − 1/T1.

  • Writing HM as (a + b)/2ab or 2(a + b)/ab.

    The formula is memorised as a jumble of the same symbols.

    Fix: Remember HM as 2ab ÷ (a + b), which is 2 over the sum of reciprocals. Check with a = b: it must give a.

  • Using GM² = AM × HM for three or more numbers.

    Students overgeneralise the two-number result.

    Fix: Use this relation only for two positive numbers. The inequality AM ≥ GM ≥ HM holds more generally for positive numbers.

  • Assuming AM > GM > HM always.

    The strict sign is remembered instead of ≥.

    Fix: Equality holds when all the numbers are equal. If the numbers are the same, all three means are equal.

Worked examples

Example 1

The 3rd term of an HP is 1/5 and the 6th term is 1/11. What is the 9th term? (a) 1/15 (b) 1/17 (c) 1/16 (d) 1/18

Show the solution
  1. Take reciprocals. The AP has 3rd term 5 and 6th term 11.
  2. Common difference d = (11 − 5) ÷ (6 − 3) = 6 ÷ 3 = 2.
  3. 9th term of the AP = 11 + 3 × 2 = 17.
  4. Flip it back. The 9th term of the HP = 1/17.

Answer: (b) 1/17

Example 2

The AM of two positive numbers is 10 and their GM is 8. What is their HM? (a) 6.4 (b) 6 (c) 9 (d) 12.5

Show the solution
  1. Use GM² = AM × HM.
  2. GM² = 64.
  3. HM = 64 ÷ 10 = 6.4.
  4. Check: 10 ≥ 8 ≥ 6.4, so the order is correct.

Answer: (a) 6.4

Example 3

What is the harmonic mean of 4 and 12? (a) 6 (b) 8 (c) 7.5 (d) 9

Show the solution
  1. HM = 2ab ÷ (a + b) with a = 4 and b = 12.
  2. 2ab = 2 × 4 × 12 = 96.
  3. a + b = 16.
  4. HM = 96 ÷ 16 = 6.
  5. Check: AM = 8, GM = √48 ≈ 6.93, HM = 6, so AM ≥ GM ≥ HM holds.

Answer: (a) 6

Exam tips

  • Most questions ask for the nth term of an HP, the HM of two numbers, or a missing mean from GM² = AM × HM. Practise these three patterns.
  • Use AM ≥ GM ≥ HM to remove options. Any option with HM above GM for unequal positive numbers can be discarded.
  • Insertion of harmonic means between two numbers means inserting arithmetic means between their reciprocals. Do it in that order.
  • Do a quick check with a = b. If your formula does not give a, it is wrong.
  • Skip lengthy HP questions on the first pass; they carry the same marks as short ones.

Practice questions from Sequence and Series

Harmonic Progression and Relation among AM, GM, HM: frequently asked questions

What is the difference between AP and HP?

In an AP, consecutive terms differ by a constant. In an HP, the reciprocals of the terms differ by a constant. So every HP question is solved by converting to an AP.

Is there a formula for the sum of an HP?

No simple formula exists, so the exam does not expect one. Questions focus on the nth term, the harmonic mean and the relation among the means.

When is AM equal to GM equal to HM?

They are equal only when all the numbers are equal and positive. For unequal positive numbers, AM is greater than GM, and GM is greater than HM.

Does GM² = AM × HM work for three numbers?

Do not use it for three numbers. It is a result for two positive numbers only, where HM = 2ab/(a+b).