Harmonic Progression, and the Three Means

• Flip every given term, find the AP's first term and common difference, find the term you need in the AP, then flip it back.
• The nth term of an HP is $\dfrac{1}{a + (n - 1)d}$, where $a$ and $d$ belong to the AP of reciprocals.
• Given two terms of an HP, flip both and use the two-term method from the Arithmetic Progression page.
• 'The common difference of the HP' in a question means the common difference of the reciprocals. An HP has no common difference of its own.
• No term of an HP can be zero.

• $H = \dfrac{2ab}{a + b}$, from $\dfrac 2H = \dfrac 1a + \dfrac 1b$.
• For unit fractions $\tfrac 1p$ and $\tfrac 1q$ it is quicker to flip: the reciprocals $p$ and $q$ have mean $\tfrac{p + q}{2}$, so $H = \dfrac{2}{p + q}$.
• 'Find $y$ if $x, y, z$ are in HP' asks for the harmonic mean of $x$ and $z$.
• The harmonic mean lies between the two numbers, nearer the smaller one.

To show a condition means an HP, clear the fractions and aim for $b(a + c) = 2ac$. If the condition factorises, look at every factor: a factor such as $a + c - b$ can make the condition true with no progression at all. When the AP and GP tests both pass, the numbers are equal: $2b = a + c$ and $b^2 = ac$ together force $(a - c)^2 = 0$.

• $A = \dfrac{a + b}{2}$, $G = \sqrt{ab}$ and $H = \dfrac{2ab}{a + b}$ for positive $a$ and $b$.
• $A \ge G \ge H$, with equality only when $a = b$.
• $G^2 = AH$, so any two of the means give the third.
• Inserted means: if $n$ numbers are put between $a$ and $b$ to make an AP, their sum is $n \times \dfrac{a + b}{2}$.
• If the $n$ numbers make an HP instead, the sum of their reciprocals is $n \times \dfrac 12 \left(\dfrac 1a + \dfrac 1b\right)$.