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Q.
If $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are the $p^ {th }, q^ {th }, r ^{th }$ terms respectively of an A.P. then the value of $ab(p-q)+b c(q-r)+c a(r-p)$ is
Sequences and Series
Solution:
Let $x$ be the first term and $y$ be the c.d. of corresponding A.P., then
$\frac{1}{a}=x+(p-1) y\, ...... (i)$
$\frac{1}{ b }= x +( q -1) y \, ...... (ii)$
$\frac{1}{ c }= x +( r -1) y\, ..... (iii)$
Multiplying (i), (ii) and (iii) respectively by
$abc (q - r), abc (r - p), abc (p - q) $ and then adding,
we get, $bc (q - r) + ca (r - p)+ ab (p - q) = 0$