Q.Verify whether the following sequence is a H.P.: $\dfrac13, \dfrac15, \dfrac17, \dfrac19, \ldots$

A sequence $t_1, t_2, t_3, \ldots, t_n, \ldots$ (with every $t_n\ne0$) is called a Harmonic Progression (H.P.) if the reciprocals of its terms, $\dfrac1{t_1}, \dfrac1{t_2}, \dfrac1{t_3}, \ldots$, form an Arithmetic Progression. There is no direct formula for the $n$th term of an H.P. in H.P. quantities alone; instead, the standard method is to write down the reciprocal sequence, identify its first term $a$ and common difference $d$ as an A.P., use $t_n(\text{A.P.})=a+(n-1)d$, and finally take the reciprocal of that expression to recover $t_n$ of the original H.P. Because reciprocal is undefined at zero, no term of a genuine H.P. can ever be zero. This detour through the reciprocal A.P. is the only reliable way to verify whether an unfamiliar sequence is an H.P., since the original terms themselves do not show a recognisable constant pattern the way an A.P. or G.P. does.

Search terms such as "harmonic progression formula and nth term" and "HP AP GP relationship important questions" recur around the Sequences and Series chapter of the NCERT/CBSE Class 11 Mathematics curriculum, a topic tested in board exams and JEE Main alike. The reciprocal-to-AP detour explained here is the one technique competitive-exam questions on H.P. almost always require, since there is no direct H.P. term formula to fall back on.