6.9 Harmonic Progression (H.P.)

6.9 Harmonic Progression (H.P.)

A sequence is a harmonic progression if the reciprocals of its terms form an A.P. Zero cannot appear (reciprocal undefined).

General form: $\dfrac{1}{a_1}, \dfrac{1}{a_1+d}, \ldots, \dfrac{1}{a_1+(n-1)d}$.

Example 22

H.P. $\dfrac{1}{2}, \dfrac{1}{5}, \dfrac{1}{8}, \ldots$: corresponding A.P. has $a_1 = 2$, $d = 3$, so

Example 23

$4$th and $7$th H.P. terms $\dfrac{2}{13}$ and $\dfrac{2}{25}$ $\Rightarrow$ A.P. terms $\dfrac{13}{2}$, $\dfrac{25}{2}$ give $d = 2$, $a_1 = \dfrac{1}{2}$.

H.P.: $\dfrac{2}{1}, \dfrac{2}{5}, \dfrac{2}{9}, \dfrac{2}{13}, \ldots$