Concept 05 / 07Sequences and Series
Harmonic Progressions and the Arithmetic, Geometric and Harmonic Means
Harmonic progressions as reciprocals of an AP, the harmonic mean, the relations A >= G >= H and G^2 = AH for two numbers, inserting means, and deciding whether three numbers are in AP, GP or HP from a single ratio.
A sequence of non-zero numbers is a harmonic progression (HP) if the reciprocals of its terms form an AP. There is no separate formula for the sum of an HP; every HP question is solved by passing to the reciprocal AP. The th term is , where is the common difference of the reciprocals. Numbers are in HP if and only if , that is, .
The harmonic mean of and is : it is the reciprocal of the average of the reciprocals. It appears whenever rates are averaged over equal amounts: driving equal distances at and km/h gives an average speed of km/h, not . Parallel resistors and lens formulas also produce harmonic means.
The three means of two positive numbers. Let , and . Then:
, with equality throughout if and only if .
, so , , themselves form a GP.
The first inequality is . The second follows from the first applied to and , or from and .
The figure gives a geometric proof. Draw a semicircle on a diameter of length , split into pieces and at a point . The radius is . The perpendicular from to the circle has length (by similar triangles, or the intersecting chords theorem, ). Dropping a perpendicular from onto the radius through the top of that segment cuts off a length along that radius (since in the right triangle). A radius is at least as long as any half-chord, and a hypotenuse is at least as long as its projection, so . The three lengths coincide only when is the centre, that is, .
Recognising the type from one ratio. For three numbers , the expression decides it:
AP.
GP.
HP.
Each follows by cross-multiplying and comparing with , and .
Inserting harmonic means: to insert harmonic means between and , insert arithmetic means between and , then take reciprocals. The reciprocal AP has common difference .
Useful facts for three numbers in HP: if are in HP, then are also in HP (add to each to get reciprocals of , and so on). Similarly, many 'show that these are in AP/GP/HP' questions reduce to one of the three defining relations after adding or multiplying by a common quantity.
Key Takeaways:
- HP: reciprocals in AP. Always convert to the reciprocal AP.
- ; average of speeds over equal distances is the harmonic mean.
- for positive numbers, with equality iff .
- for two numbers.
- equals , or for AP, GP or HP.
- for in HP.
- Pitfall: trying to use a 'sum of HP' formula; none exists.
Brain-Teaser 1: If the AM and GM of two positive numbers are and , find the numbers and their HM.
Solution: and , so are the roots of , namely and . The HM is , which also equals . Using gives the HM without finding the numbers at all.
Brain-Teaser 2: If are in HP, show that .
Solution: From we get . Then and . So . Substituting the HP relation for the middle term and simplifying is the direct route; the cancels neatly, which is the sign that the algebra is right.
Brain-Teaser 3: A car travels from A to B at km/h and returns at km/h. Another car does the round trip at a constant km/h. Which is faster, and why?
Solution: The first car's average speed is the harmonic mean of and (equal distances): km/h. The second car averages km/h, which is the arithmetic mean. Since for unequal speeds, the second car always wins. Intuitively, the first car spends twice as long at the slow speed as at the fast one, so the slow speed dominates the average. The inequality is the precise statement of this intuition.
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