Harmonic Progression (HP)

A Harmonic Progression (HP) is a sequence whose reciprocals form an Arithmetic Progression. That is, $t_1, t_2, t_3, \ldots$ is an HP precisely when $\dfrac{1}{t_1}, \dfrac{1}{t_2}, \dfrac{1}{t_3}, \ldots$ is an AP (with no term equal to zero).

Example: the sequence $\dfrac{1}{6}, \dfrac{1}{9}, \dfrac{1}{12}, \dfrac{1}{15}, \ldots$ is an HP because its reciprocals $6, 9, 12, 15, \ldots$ form an AP with $a = 6$ and $d = 3$. To find the $10$th term of the HP: the $10$th term of the reciprocal AP is $6 + (10-1)(3) = 6 + 27 = 33$, so the $10$th term of the HP is $\dfrac{1}{33}$.