If p, q, r are in A.P. and x, y, z are in G.P., then xq−r⋅yr−p⋅zp−$$q=$$

Geometric progression

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If p, q, r are in A.P. and x, y, z are in G.P., then $x^{q - r} \cdot y^{r - p} \cdot z^{p - q} =$

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Solution

Let d be the common difference of A.P. and R (≠ 0), the common ratio of G.P., then
$q = p + d, r = p + 2 d$
and $y = xR, z = {xR}^{2}$
so that $q - r = - d, r - p = 2 d, p - q = - d$
$\begin{array}{l} ∴ x^{q - r} \cdot y^{r - p} \cdot z^{p - q} = x^{- d} \cdot (xR)^{2 d} \cdot {({xR}^{2})}^{- d} \\ = (x^{- d} \cdot x^{2 d} \cdot x^{- d}) (R^{2 d} \times R^{- 2 d}) \\ = (x^{- d + 2 d - d}) \cdot (R^{2 d - 2 d}) \\ = x^{0} \cdot R^{0} = 1 \times 1 = 1 \end{array}$

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Geometric progression

Remember concepts with our Masterclasses.

If p, q, r are in A.P. and x, y, z are in G.P., then xq−r⋅yr−p⋅zp−q=

Let d be the common difference of A.P. and R (≠ 0), the common ratio of G.P., thenq=p+d,r=p+2dand y=xR,z=xR2so that q−r=−d,r−p=2d,p−q=−d∴ xq−r⋅yr−p⋅zp−q=x−d⋅(xR)2d⋅xR2−d=x−d⋅x2d⋅x−dR2d×R−2d=x−d+2d−d⋅R2d−2d=x0⋅R0=1×1=1

Ready to Test Your Skills?

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If p, q, r are in A.P. and x, y, z are in G.P., then $x^{q - r} \cdot y^{r - p} \cdot z^{p - q} =$