document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151135>Question 1151135</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Three numbers form a geometric progression. If 4 is subtracted from the third term, then the three numbers will form an arithmetic progression. If, after this, 1 is subtracted from the second and third terms of the progression, then it will again result in a geometric progression. Find these three numbers.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #772775 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13198)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=772775\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "There are probably nicer ways to solve this problem....  But this is what I came up with.<br><BR>\n" );
document.write( "The original geometric progression is<br><BR>\n" );
document.write( "a, ar, ar^2<br><BR>\n" );
document.write( "When 4 is subtracted from the third number, the resulting progression<br><BR>\n" );
document.write( "a, ar, ar^2-4<br><BR>\n" );
document.write( "is an arithmetic progression.<br><BR>\n" );
document.write( "Then when 1 is subtracted from the second and third terms, the resulting progression<br><BR>\n" );
document.write( "a, ar-1, ar^2-5<br><BR>\n" );
document.write( "is again a geometric progression.<br><BR>\n" );
document.write( "(1) Using the fact that the last progression is geometric:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ar%5E2-5%29%2F%28ar-1%29+=+%28ar-1%29%2Fa ALIGN=MIDDLE  ALT=\"%28ar%5E2-5%29%2F%28ar-1%29+=+%28ar-1%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2r%5E2-2ar%2B1+=+a%5E2r%5E2-5a ALIGN=MIDDLE  ALT=\"a%5E2r%5E2-2ar%2B1+=+a%5E2r%5E2-5a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2ar%2B1+=+-5a ALIGN=MIDDLE  ALT=\"-2ar%2B1+=+-5a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+2ar-5a ALIGN=MIDDLE  ALT=\"1+=+2ar-5a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+a%282r-5%29 ALIGN=MIDDLE  ALT=\"1+=+a%282r-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+1%2F%282r-5%29 ALIGN=MIDDLE  ALT=\"a+=+1%2F%282r-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > [1]<br><BR>\n" );
document.write( "(2) Using the fact that the second progression is arithmetic:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ar%5E2-4%29-ar+=+ar-a ALIGN=MIDDLE  ALT=\"%28ar%5E2-4%29-ar+=+ar-a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ar%5E2-2ar%2Ba+=+4 ALIGN=MIDDLE  ALT=\"ar%5E2-2ar%2Ba+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > [2]<br><BR>\n" );
document.write( "(3) Substituting [2] in [1]....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2%2F%282r-5%29-2r%2F%282r-5%29%2B1%2F%282r-5%29+=+4 ALIGN=MIDDLE  ALT=\"r%5E2%2F%282r-5%29-2r%2F%282r-5%29%2B1%2F%282r-5%29+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2-2r%2B1+=+4%282r-5%29 ALIGN=MIDDLE  ALT=\"r%5E2-2r%2B1+=+4%282r-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2-2r%2B1+=+8r-20 ALIGN=MIDDLE  ALT=\"r%5E2-2r%2B1+=+8r-20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2-10r%2B21+=+0 ALIGN=MIDDLE  ALT=\"r%5E2-10r%2B21+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r-3%29%28r-7%29+=+0 ALIGN=MIDDLE  ALT=\"%28r-3%29%28r-7%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Both solutions satisfy the conditions of the problem.<br><BR>\n" );
document.write( "(A) r = 3<br><BR>\n" );
document.write( "From [1], a = 1/(2(3)-5) = 1/1 = 1.<br><BR>\n" );
document.write( "The original sequence is<br><BR>\n" );
document.write( "1, 3, 9<br><BR>\n" );
document.write( "When 4 is subtracted from the third number, the resulting sequence is<br><BR>\n" );
document.write( "1, 3, 5<br><BR>\n" );
document.write( "which is an arithmetic progression.<br><BR>\n" );
document.write( "Then when 1 is subtracted from each of the second and third terms, the resulting sequence is<br><BR>\n" );
document.write( "1, 2, 4<br><BR>\n" );
document.write( "which is again a geometric progression.<br><BR>\n" );
document.write( "(B) r = 7<br><BR>\n" );
document.write( "From [1], a = 1/(2(7)-5) = 1/9.<br><BR>\n" );
document.write( "The original sequence is<br><BR>\n" );
document.write( "1/9, 7/9, 49/9<br><BR>\n" );
document.write( "When 4 is subtracted from the third number, the resulting sequence is<br><BR>\n" );
document.write( "1/9, 7/9, 13/9<br><BR>\n" );
document.write( "which is an arithmetic progression.<br><BR>\n" );
document.write( "Then when 1 is subtracted from each of the second and third terms, the resulting sequence is<br><BR>\n" );
document.write( "1/9, -2/9, 4/9<br><BR>\n" );
document.write( "which is again a geometric progression.<br>\r<BR>\n" );
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