A harmonic sequence is an arithmetic one wearing a disguise — flip every term and the familiar rules take over.
The problem
What is the next term of 12, 15, 18, 111, … ?
Flip the terms: 2, 5, 8, 11 — arithmetic with d = 3, so the next reciprocal is 14 and the next term is 114. A adds 2 instead of 3; nothing about the fractions themselves changes by a constant — only their reciprocals do.
harmonic: 12, 15, 18, 111, …
the sequence as given
flipped: 2, 5, 8, 11, …
arithmetic — common difference 3
That flip is the entire theory. Every harmonic question is an arithmetic question plus two flips — one on the way in, one on the way out.
Find the 10th term of the sequence above.
flipped: a = 2, d = 3
work with the reciprocals
a₁₀ = 2 + 9(3) = 29
nine jumps to term ten
10th harmonic term = 129
flip back at the end
Which term is it? Which term of 12, 15, 18, … is 144?
44 = 2 + (n − 1)(3)
flip 144 to 44 first
n − 1 = 14, n = 15
the 15th term
Two terms given. The 2nd term is 19 and the 6th is 121. Find the 8th.
flipped: a₂ = 9, a₆ = 21
an arithmetic problem now
d = (21 − 9) ÷ (6 − 2) = 3
gap in values ÷ gap in positions
a₈ = 21 + 2(3) = 27, so the 8th term is 127
flip back
HM = 2aba + b
the middle term between a and b, harmonically
HM of 3 and 6 = 2 · 189 = 4
twice the product over the sum: 2(3)(6) = 36, 3 + 6 = 9
Check the definition: 3, 4, 6 flips to 13, 14, 16 — differences −112 and −112 — arithmetic ✓. The harmonic mean also runs the famous round-trip average-speed shortcut: going out at 30 km/h and back over the same road at 60 km/h averages 2(30)(60) ÷ (30 + 60) = 40 km/h, not the 45 you get by averaging the speeds.
| Temptation | Why it fails |
|---|---|
| subtract neighbouring terms | 15 − 12 ≠ 18 − 15 — no common difference here |
| average a and b for the mean | 3 + 62 = 4.5 is the arithmetic mean, not 4 |
| hunt for a sum formula | harmonic series have no clean sum — exams do not ask for one |
flip: 3, __, __, __, 15
five terms, four gaps
d = (15 − 3) ÷ 4 = 3
arithmetic means 6, 9, 12
flip back: 16, 19, 112
the harmonic means
If the question asks for their sum, add the fractions themselves: 16 + 19 + 112 = 636 + 436 + 336 = 1336.
6 + 9 + 12 = 27 does not make the sum 127. Flipping works one term at a time, never on a sum.
The terms are not 1 over something. Find the 5th term of the harmonic sequence 6, 3, 2, …
flip: 16, 13, 12, …
d = 13 − 16 = 16
5th flipped term = 16 + 4(16) = 56
four jumps
5th term = 65
flip back
What is different: a whole number flips to a fraction, so the arithmetic sequence is the one with fractions. The method is the same.
The reciprocals go down. Find the 4th term of 17, 14, 1, …
flip: 7, 4, 1, …
d = −3
4th flipped term = 1 − 3 = −2
past zero
4th term = −12
flip back, sign and all
What is different: the harmonic terms grow, then jump to a negative, and the sign carries through the flip. A flipped term of 0 would have no reciprocal, so 0 can never sit in the arithmetic list.
Fractions for a and b. Find the harmonic mean of 12 and 13.
flip: 2 and 3, their average is 52
the arithmetic mean of the flips
HM = 25
flip back
2(12)(13) ÷ (12 + 13) = (13) ÷ (56) = 25 ✓
the formula agrees
What is different: with fractions, flipping first is quicker than the formula, and it lands in the same place.
For positive a and b, AM × HM = a + b2 × 2aba + b = ab, and ab is GM².
2 and 8: AM = 5, HM = 2(2)(8)10 = 3.2, GM = 4
test it on two numbers
5 × 3.2 = 16 = 4² ✓
the link holds
GM² = 10 × 6.4 = 64
now the question
GM = 8
positive numbers, positive root
The numbers themselves are 4 and 16: sum 20, product 64, and 2(64) ÷ 20 = 6.4 ✓. The GM always sits between the HM and the AM.
64 is GM², the product of the numbers. Take the square root.
Harmonic items turn into arithmetic ones once you flip: the nth term, which term is 1k, several means inserted, or the round-trip average speed. The classic trap is averaging or adding the fractions as if they were the arithmetic terms. Ten-second check: flip your answer and the reciprocals must step by the same difference.
The problem
What is the harmonic mean of 4 and 12?
HM = 2aba + b = 2(4)1216 = 9616 = 6. A is the arithmetic mean and B (≈6.93) the geometric mean — three different means, and the harmonic one is always the smallest of the three. Check: 4, 6, 12 flips to 14, 16, 112 with equal steps of −112 ✓.
