document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177109>Question 1177109</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a quadratic pattern has a second term equal to 1, a third term equal to -6 and fifth term equal to -14\r<BR>\n" );
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document.write( "1.1.1 calculate the second difference of this quadratic pattern \r<BR>\n" );
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document.write( "1.1.2 hence or otherwise, calculate the first term of the pattern\r<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 #805357 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</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=805357\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "A quadratic sequence has a constant second difference.  So this is what we know about the sequence:<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "  ___    1    -6    ___   -14    terms 1 through 5\r\n" );
document.write( "     ___   -7    ___   ___       first differences\r\n" );
document.write( "         x     x     x           constant second differences<br></PRE><BR>\n" );
document.write( "We can get expressions for the other first differences using the one known first difference and the constant second differences:<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "  ___     1    -6     ___     -14    terms 1 through 5\r\n" );
document.write( "     -7-x   -7    -7+x   -7+2x       first differences\r\n" );
document.write( "          x     x      x             constant second differences<br></PRE><BR>\n" );
document.write( "Now we can get two different expressions in x for the 4th term of the sequence, using the 3rd and 5th terms and the second differences.  Find the value of x that makes both expressions the same.<br><BR>\n" );
document.write( "4th term, using the 3rd term -6 and the common difference (-7+x): -13+x<BR>\n" );
document.write( "4th term, using the 5th term -14 and the common difference (-7+2x): -7-2x<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-13%2Bx+=+-7-2x ALIGN=MIDDLE  ALT=\"-13%2Bx+=+-7-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x+=+6 ALIGN=MIDDLE  ALT=\"3x+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2 ALIGN=MIDDLE  ALT=\"x+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we can use that value of x to fill in the array, working from the bottom up:<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "   8    1   -6   -11   -14    terms 1 through 5\r\n" );
document.write( "     -9   -7   -5    -3       first differences\r\n" );
document.write( "        2    2     2          constant second differences<br></PRE><BR>\n" );
document.write( "ANSWERS: The constant second difference is 2; the first term is 8.<br><BR>\n" );
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