document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176668>Question 1176668</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A sequence of positive integers with a_1 = 1 and a_9 + a_{10} = 646 is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all i>=1, the terms a_{2i - 1}, a_{2i}, a_{2i + 1} are in geometric progression, and the terms a_{2i}, a_{2i + 1}, and a_{2i + 2} are in arithmetic progression. Find the greatest term in this sequence that is less than 1000.<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 #804011 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</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=804011\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "A formal algebraic solution would probably be very messy....<br><BR>\n" );
document.write( "But since the sequence is all integers, we can find the sequence we are looking for simply by investigation.<br><BR>\n" );
document.write( "Second term 2: 1, 2, 4, 6, 9, 12, 16, 20, 25, 30;  sum 55<BR>\n" );
document.write( "Second term 3: 1, 3, 9, 15, 25, 35, 49, 63, 81, 109;  sum 190<BR>\n" );
document.write( "Second term 4: 1, 4, 16, 28, 49, 70, 100, 130, 169, 208;  sum 377<BR>\n" );
document.write( "Second term 5: 1, 5, 25, 45, 81, 117, 169, 221, 289, 357; sum 646<br><BR>\n" );
document.write( "That is the sequence we want; continue the sequence until the numbers exceed 1000.<br><BR>\n" );
document.write( "1, 5, 25, 45, 81, 117, 169, 221, 289, 357, 441, 525, 625, 725, 841, 957, 1089<br><BR>\n" );
document.write( "The largest number in the sequence less than 1000 is 957.<br><BR>\n" );
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