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Q.
If $p, q, r $ are in A.P. and $x, y, z$ are in G.P. then $x^{q-r} . y ^{r-p}.z^{p-q}$ is equal to
Sequences and Series
Solution:
Since $p, q,r$ are in $A.P$.
$\therefore 2q= p+r\quad...\left(i\right)$
Since $x, y, z$ are in $G.P$.
Let $y= xR, z=xR^{2}$, where $R$ is the ratio of the $G.P$.
$ \therefore x^{q-r}\left(xR\right)^{r-p} \left(xR^{2}\right)^{p-q} = x^{q-r+r-p+p-q} R^{r-p+2p-2q} $
$ = x^{0}R^{r+p-2q} = x^{0}R^{0} = 1\cdot1 = 1 \quad$ [By $\left(1\right)$ ]