document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=711100>Question 711100</A>: <I><SPAN STYLE=\"word-break: break-all;\"> THREE INTEGERS ARE IN A GEOMETRIC SERIES, WHEN 12 IS ADDED TO THE SECOND TERM THE SERIES BECOMES AN ARITHMETIC SERIES  AND WHEN 96 IS ALSO ADDED TO THE THIRD TERM THE SERIES CHANGES TO GEOMETRIC SERIES. WHAT ARE THE INTEGERS?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #438816 by </B><A HREF=/tutors/aboutme.mpl?userid=kevwill><B>kevwill(135)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=kevwill><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=kevwill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=438816\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's call the three integers in the original series a, b, and c.  Because this is a geometric series, we know that<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Fa+=+c%2Fb ALIGN=MIDDLE  ALT=\"b%2Fa+=+c%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so multiplying both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Ab ALIGN=MIDDLE  ALT=\"a%2Ab\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+=+ac ALIGN=MIDDLE  ALT=\"b%5E2+=+ac\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+b%5E2%2Fa ALIGN=MIDDLE  ALT=\"c+=+b%5E2%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Adding 12 to the second number (b) gives an arithmetic series, so we know that<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2B12+-+a+=+c+-+%28b%2B12%29 ALIGN=MIDDLE  ALT=\"b%2B12+-+a+=+c+-+%28b%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+c+=+2b%2B24 ALIGN=MIDDLE  ALT=\"a+%2B+c+=+2b%2B24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "And finally, adding 96 to the third term of the new series gives us another geometric series, so we have<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B12%29%2Fa+=+%28c%2B96%29%2F%28b%2B12%29 ALIGN=MIDDLE  ALT=\"%28b%2B12%29%2Fa+=+%28c%2B96%29%2F%28b%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so multiplying both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A%28b%2B12%29 ALIGN=MIDDLE  ALT=\"a%2A%28b%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B12%29%5E2+=+a%2A%28c%2B96%29 ALIGN=MIDDLE  ALT=\"%28b%2B12%29%5E2+=+a%2A%28c%2B96%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+%2B+24b+%2B+144+=+ac+%2B+96a ALIGN=MIDDLE  ALT=\"b%5E2+%2B+24b+%2B+144+=+ac+%2B+96a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+=+ac ALIGN=MIDDLE  ALT=\"b%5E2+=+ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24b+%2B+144+=+96a ALIGN=MIDDLE  ALT=\"24b+%2B+144+=+96a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+4a-6 ALIGN=MIDDLE  ALT=\"b+=+4a-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "And using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+b%5E2%2Fa ALIGN=MIDDLE  ALT=\"c+=+b%5E2%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+%284a-6%29%5E2%2Fa ALIGN=MIDDLE  ALT=\"c+=+%284a-6%29%5E2%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+%2816a%5E2-48a%2B36%29%2Fa ALIGN=MIDDLE  ALT=\"c+=+%2816a%5E2-48a%2B36%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=c+=+16a+-+48+%2B+36%2Fa ALIGN=MIDDLE  ALT=\"c+=+16a+-+48+%2B+36%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "From the equation above, c is only an integer when a is a factor if 36, so we can check the nine possibilities:<BR>\n" );
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document.write( "(1, -2, 4)<BR>\n" );
document.write( "(2, 2, 2)<BR>\n" );
document.write( "(3, 6, 12)<BR>\n" );
document.write( "(4, 10, 25)<BR>\n" );
document.write( "(6, 18, 54)<BR>\n" );
document.write( "(9, 30, 100)<BR>\n" );
document.write( "(12, 42, 147)<BR>\n" );
document.write( "(18, 66, 242)<BR>\n" );
document.write( "(36, 138, 529)<BR>\n" );
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document.write( "Of these 9 geometric series, only one generates an arithmetic series when 12 is added to the second term:  (6, 18, 54)<BR>\n" );
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document.write( "Adding 12 to the second term gives the arithmetic series (6, 30, 54), <BR>\n" );
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document.write( "And adding 96 to the third term gives the geometric series (6, 30, 150).<BR>\n" );
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document.write( "  </SPAN>\n" );
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