document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184306>Question 1184306</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let a, b, c, p, q, r be positive real numbers such that a, b, c are in geometric sequence and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5Ep+=+b%5Eq+=+c%5Er+ ALIGN=MIDDLE  ALT=\"+a%5Ep+=+b%5Eq+=+c%5Er+\"   DISABLED_style= \"max-width:90%; height:auto;\" > then which one of the following condition holds:<BR>\n" );
document.write( "A- p, q, r are in geometric sequence<BR>\n" );
document.write( "B- p, q, r are in arithmetic sequence<BR>\n" );
document.write( "C- p, q, r are in harmonic sequence<BR>\n" );
document.write( "D- p^2, q^2, r^2 are in arithmetic sequence<BR>\n" );
document.write( "E- p^2, q^2, r^2 are geometric sequence<BR>\n" );
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document.write( "[Note: ^2 means power 2] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849857 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849857\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to determine the correct relationship between p, q, and r:\r<BR>\n" );
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document.write( "1. **Express the geometric sequence:**\r<BR>\n" );
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document.write( "Since a, b, and c are in a geometric sequence, we can write $b^2 = ac$.\r<BR>\n" );
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document.write( "2. **Express the given relationship:**\r<BR>\n" );
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document.write( "We are given that $ap = bq = cr$. Let's call this common value k. Then:<BR>\n" );
document.write( "* $ap = k \Rightarrow a = \frac{k}{p}$<BR>\n" );
document.write( "* $bq = k \Rightarrow b = \frac{k}{q}$<BR>\n" );
document.write( "* $cr = k \Rightarrow c = \frac{k}{r}$\r<BR>\n" );
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document.write( "3. **Substitute into the geometric sequence equation:**\r<BR>\n" );
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document.write( "Substitute the expressions for a, b, and c into the equation $b^2 = ac$:\r<BR>\n" );
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document.write( "$(\frac{k}{q})^2 = (\frac{k}{p})(\frac{k}{r})$\r<BR>\n" );
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document.write( "$\frac{k^2}{q^2} = \frac{k^2}{pr}$\r<BR>\n" );
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document.write( "4. **Simplify:**\r<BR>\n" );
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document.write( "Since k is a positive real number, we can divide both sides by $k^2$:\r<BR>\n" );
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document.write( "$\frac{1}{q^2} = \frac{1}{pr}$\r<BR>\n" );
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document.write( "$q^2 = pr$\r<BR>\n" );
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document.write( "5. **Interpret the result:**\r<BR>\n" );
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document.write( "The equation $q^2 = pr$ means that p, q, and r are in a geometric sequence.\r<BR>\n" );
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document.write( "Therefore, the correct condition is that p, q, and r are in geometric sequence.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{A}$<BR>\n" );
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