Q.Find the $n$th term and hence find the 8th term of the H.P.: $\dfrac12, \dfrac15, \dfrac18, \dfrac1{11}, \ldots$

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. …