Harmonic Progression (H.P.)

A sequence $t_1,t_2,t_3,\ldots,t_n,\ldots$ (with $t_n\ne0$ for all $n\in\mathbb N$) is called a Harmonic Progression (H.P.) if the reciprocals $\dfrac1{t_1},\dfrac1{t_2},\dfrac1{t_3},\ldots,\dfrac1{t_n},\ldots$ are in A.P. E.g. (i) $\dfrac17,\dfrac1{11},\dfrac1{15},\ldots$ are in H.P. because $7,11,15,\ldots$ (their reciprocals) are in A.P. (ii) $\dfrac14,\dfrac3{14},\dfrac3{16},\ldots$ is H.P. because $4,\dfrac{14}3,\dfrac{16}3,\ldots$ are in A.P.

Worked Example 1: find the $n$th term of the H.P. $\dfrac12,\dfrac25,\dfrac13,\dfrac27,\ldots$. The reciprocals $2,\dfrac52,3,\dfrac72,\ldots$ are in A.P. with $a=2, d=\dfrac12$, so their $n$th term is $t_n(\text{A.P.})=2+(n-1)\dfrac12=\dfrac{3+n}2$; hence for the H.P. $t_n=\dfrac2{3+n}$. …