If P equiv left( frac{1}{x_p}, p right), Q = | vedclass

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(C) Let the $k^{th}$ term of the $H.P.$ be $x_k$. By definition,$\frac{1}{x_k}$ is the $k^{th}$ term of an $A.P.$Let $a_k = \frac{1}{x_k} = a + (k-1)d

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the points $P, Q, R$ are collinear

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(C) Let the $k^{th}$ term of the $H.P.$ be $x_k$. By definition,$\frac{1}{x_k}$ is the $k^{th}$ term of an $A.P.$Let $a_k = \frac{1}{x_k} = a + (k-1)d$,where $a$ is the first term and $d$ is the common difference.The coordinates of the points are $P(a_p, p)$,$Q(a_q, q)$,and $R(a_r, r)$.The slope of line segment $PQ$ is $m_{PQ} = \frac{q - p}{a_q - a_p} = \frac{q - p}{(a + (q-1)d) - (a + (p-1)d)} = \frac{q - p}{(q - p)d} = \frac{1}{d}$.The slope of line segment $QR$ is $m_{QR} = \frac{r - q}{a_r - a_q} = \frac{r - q}{(a + (r-1)d) - (a + (q-1)d)} = \frac{r - q}{(r - q)d} = \frac{1}{d}$.Since the slopes $m_{PQ} = m_{QR} = \frac{1}{d}$,the points $P, Q,$ and $R$ lie on the same line.Therefore,the points $P, Q, R$ are collinear.

If $P \equiv \left( \frac{1}{x_p}, p \right), Q = \left( \frac{1}{x_q}, q \right), R = \left( \frac{1}{x_r}, r \right)$ where $x_k \neq 0$ denotes the $k^{th}$ term of an $H.P.$ for $k \in N$,then:

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