document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209773>Question 1209773</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a_1, a_2, ..., a_19 satisfy\r<BR>\n" );
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document.write( "a_1 + a_2 + a_3 + a_4 = 1, <BR>\n" );
document.write( "a_2 + a_3 + a_4 + a_5 = 2, <BR>\n" );
document.write( "a_3 + a_4 + a_5 + a_6 = 3, \r<BR>\n" );
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document.write( "...\r<BR>\n" );
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document.write( "a_{16} + a_{17} + a_{18} + a_{19} = 16,<BR>\n" );
document.write( "a_{17} + a_{18} + a_{19} + a_1 = 17,\r<BR>\n" );
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document.write( "a_{18} + a_{19} + a_1 + a_2 = 18,\r<BR>\n" );
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document.write( "a_{19} + a_1 + a_2 + a_3 = 19,\r<BR>\n" );
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document.write( "what is the value of a_19?<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 #850326 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=850326\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze the given system of equations.\r<BR>\n" );
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document.write( "We have:<BR>\n" );
document.write( "1.  a_1 + a_2 + a_3 + a_4 = 1<BR>\n" );
document.write( "2.  a_2 + a_3 + a_4 + a_5 = 2<BR>\n" );
document.write( "3.  a_3 + a_4 + a_5 + a_6 = 3<BR>\n" );
document.write( "...<BR>\n" );
document.write( "16. a_16 + a_17 + a_18 + a_19 = 16<BR>\n" );
document.write( "17. a_17 + a_18 + a_19 + a_1 = 17<BR>\n" );
document.write( "18. a_18 + a_19 + a_1 + a_2 = 18<BR>\n" );
document.write( "19. a_19 + a_1 + a_2 + a_3 = 19\r<BR>\n" );
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document.write( "Subtract equation (1) from equation (2):<BR>\n" );
document.write( "(a_2 + a_3 + a_4 + a_5) - (a_1 + a_2 + a_3 + a_4) = 2 - 1<BR>\n" );
document.write( "a_5 - a_1 = 1<BR>\n" );
document.write( "a_5 = a_1 + 1\r<BR>\n" );
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document.write( "Subtract equation (2) from equation (3):<BR>\n" );
document.write( "(a_3 + a_4 + a_5 + a_6) - (a_2 + a_3 + a_4 + a_5) = 3 - 2<BR>\n" );
document.write( "a_6 - a_2 = 1<BR>\n" );
document.write( "a_6 = a_2 + 1\r<BR>\n" );
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document.write( "In general, we have a_{n+4} = a_n + 1.\r<BR>\n" );
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document.write( "This means:<BR>\n" );
document.write( "a_5 = a_1 + 1<BR>\n" );
document.write( "a_9 = a_5 + 1 = a_1 + 2<BR>\n" );
document.write( "a_13 = a_9 + 1 = a_1 + 3<BR>\n" );
document.write( "a_17 = a_13 + 1 = a_1 + 4\r<BR>\n" );
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document.write( "Similarly,<BR>\n" );
document.write( "a_6 = a_2 + 1<BR>\n" );
document.write( "a_10 = a_6 + 1 = a_2 + 2<BR>\n" );
document.write( "a_14 = a_10 + 1 = a_2 + 3<BR>\n" );
document.write( "a_18 = a_14 + 1 = a_2 + 4\r<BR>\n" );
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document.write( "Also,<BR>\n" );
document.write( "a_7 = a_3 + 1<BR>\n" );
document.write( "a_11 = a_7 + 1 = a_3 + 2<BR>\n" );
document.write( "a_15 = a_11 + 1 = a_3 + 3<BR>\n" );
document.write( "a_19 = a_15 + 1 = a_3 + 4\r<BR>\n" );
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document.write( "And,<BR>\n" );
document.write( "a_8 = a_4 + 1<BR>\n" );
document.write( "a_12 = a_8 + 1 = a_4 + 2<BR>\n" );
document.write( "a_16 = a_12 + 1 = a_4 + 3\r<BR>\n" );
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document.write( "Now let's use equations 1, 17, 18, and 19.<BR>\n" );
document.write( "1. a_1 + a_2 + a_3 + a_4 = 1<BR>\n" );
document.write( "17. a_17 + a_18 + a_19 + a_1 = 17<BR>\n" );
document.write( "18. a_18 + a_19 + a_1 + a_2 = 18<BR>\n" );
document.write( "19. a_19 + a_1 + a_2 + a_3 = 19\r<BR>\n" );
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document.write( "Substituting the expressions we found:<BR>\n" );
document.write( "17. (a_1 + 4) + (a_2 + 4) + (a_3 + 4) + a_1 = 17<BR>\n" );
document.write( "2a_1 + a_2 + a_3 + 12 = 17<BR>\n" );
document.write( "2a_1 + a_2 + a_3 = 5\r<BR>\n" );
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document.write( "18. (a_2 + 4) + (a_3 + 4) + a_1 + a_2 = 18<BR>\n" );
document.write( "a_1 + 2a_2 + a_3 + 8 = 18<BR>\n" );
document.write( "a_1 + 2a_2 + a_3 = 10\r<BR>\n" );
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document.write( "19. (a_3 + 4) + a_1 + a_2 + a_3 = 19<BR>\n" );
document.write( "a_1 + a_2 + 2a_3 + 4 = 19<BR>\n" );
document.write( "a_1 + a_2 + 2a_3 = 15\r<BR>\n" );
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document.write( "From 1, a_4 = 1 - (a_1 + a_2 + a_3).<BR>\n" );
document.write( "From 19, a_1 + a_2 + a_3 = 19 - a_19.<BR>\n" );
document.write( "Substitute into 1:<BR>\n" );
document.write( "19 - a_19 + a_4 = 1, so a_4 = a_19 - 18.<BR>\n" );
document.write( "Since a_19 = a_3 + 4, a_3 = a_19 - 4.<BR>\n" );
document.write( "From 1, a_1 + a_2 + a_3 = 1 - a_4 = 1 - (a_19 - 18) = 19 - a_19.<BR>\n" );
document.write( "Now substitute a_3 = a_19 - 4 into a_1 + a_2 + 2a_3 = 15.<BR>\n" );
document.write( "a_1 + a_2 + 2(a_19 - 4) = 15<BR>\n" );
document.write( "a_1 + a_2 + 2a_19 - 8 = 15<BR>\n" );
document.write( "a_1 + a_2 = 23 - 2a_19.<BR>\n" );
document.write( "Substitute into a_1 + a_2 + a_3 = 19 - a_19.<BR>\n" );
document.write( "23 - 2a_19 + a_19 - 4 = 19<BR>\n" );
document.write( "19 - a_19 = 19<BR>\n" );
document.write( "a_19 = 0.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{0}$<BR>\n" );
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