document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=970169>Question 970169</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What are the next two numbers in this series 1,8,27,58<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 #592825 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=592825\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27 ALIGN=MIDDLE  ALT=\"27\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=58 ALIGN=MIDDLE  ALT=\"58\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "first find the pattern:\r<BR>\n" );
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document.write( "To find the pattern, I will list the numbers, and find the differences for each pair of numbers. That is, I will subtract the numbers in pairs (the first from the second, the second from the third, and so on), like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >..........<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27 ALIGN=MIDDLE  ALT=\"27\"   DISABLED_style= \"max-width:90%; height:auto;\" >..........<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=58 ALIGN=MIDDLE  ALT=\"58\"   DISABLED_style= \"max-width:90%; height:auto;\" >.....\r<BR>\n" );
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document.write( ".....<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19 ALIGN=MIDDLE  ALT=\"19\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=31 ALIGN=MIDDLE  ALT=\"31\"   DISABLED_style= \"max-width:90%; height:auto;\" >.... the \"first differences\", are not all the same value, I'll continue subtracting:<BR>\n" );
document.write( "..........<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" >....=> the second differences are the same, so the polynomial for this sequence of values is a quadratic<BR>\n" );
document.write( "Since the formula for the terms is a quadratic, then I know that it is of the form:\r<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=an%5E2+%2B+bn+%2B+c+ ALIGN=MIDDLE  ALT=\"an%5E2+%2B+bn+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now I have to find those numbers by plugging in some of the values from the sequence, and then solving the resulting system of equations. \r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=1 ALIGN=MIDDLE  ALT=\"n=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > first term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A1%5E2+%2B+b%2A1+%2B+c=1+ ALIGN=MIDDLE  ALT=\"a%2A1%5E2+%2B+b%2A1+%2B+c=1+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+b+%2B+c=1 ALIGN=MIDDLE  ALT=\"a+%2B+b+%2B+c=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ..........eq.1\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=2 ALIGN=MIDDLE  ALT=\"n=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > second term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A2%5E2+%2B+b%2A2+%2B+c=8+ ALIGN=MIDDLE  ALT=\"a%2A2%5E2+%2B+b%2A2+%2B+c=8+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a+%2B+2b+%2B+c=8 ALIGN=MIDDLE  ALT=\"4a+%2B+2b+%2B+c=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > ..........eq.2\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=3 ALIGN=MIDDLE  ALT=\"n=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > third term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27 ALIGN=MIDDLE  ALT=\"27\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A3%5E2+%2B+b%2A3+%2B+c=27 ALIGN=MIDDLE  ALT=\"a%2A3%5E2+%2B+b%2A3+%2B+c=27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9a+%2B+3b+%2B+c=27 ALIGN=MIDDLE  ALT=\"9a+%2B+3b+%2B+c=27\"   DISABLED_style= \"max-width:90%; height:auto;\" > ..........eq.3\r<BR>\n" );
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document.write( "start with\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a+%2B+2b+%2B+c=8 ALIGN=MIDDLE  ALT=\"4a+%2B+2b+%2B+c=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > ..........eq.2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+b+%2B+c=1 ALIGN=MIDDLE  ALT=\"a+%2B+b+%2B+c=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ..........eq.1<BR>\n" );
document.write( "--------------------------------subtract  <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a+%2B+b=7 ALIGN=MIDDLE  ALT=\"3a+%2B+b=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > --------solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=7-3a ALIGN=MIDDLE  ALT=\"b=7-3a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........................(1)\r<BR>\n" );
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document.write( "now use <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9a+%2B+3b+%2B+c=27 ALIGN=MIDDLE  ALT=\"9a+%2B+3b+%2B+c=27\"   DISABLED_style= \"max-width:90%; height:auto;\" > ..........eq.3<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a+%2B+2b+%2B+c=8 ALIGN=MIDDLE  ALT=\"4a+%2B+2b+%2B+c=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > ..........eq.2<BR>\n" );
document.write( "--------------------------------subtract <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5a%2Bb=19 ALIGN=MIDDLE  ALT=\"5a%2Bb=19\"   DISABLED_style= \"max-width:90%; height:auto;\" >--------solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=19-5a ALIGN=MIDDLE  ALT=\"b=19-5a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........................(2)\r<BR>\n" );
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document.write( "use (1) and (2) to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7-3a=19-5a ALIGN=MIDDLE  ALT=\"7-3a=19-5a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5a-3a=19-7 ALIGN=MIDDLE  ALT=\"5a-3a=19-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a=12 ALIGN=MIDDLE  ALT=\"2a=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a=6%29 ALIGN=MIDDLE  ALT=\"highlight%28a=6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=7-3a ALIGN=MIDDLE  ALT=\"b=7-3a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........................(1)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=7-3%2A6 ALIGN=MIDDLE  ALT=\"b=7-3%2A6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=7-18 ALIGN=MIDDLE  ALT=\"b=7-18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b=-11%29 ALIGN=MIDDLE  ALT=\"highlight%28b=-11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "go to eq.1 substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+b+%2B+c=1 ALIGN=MIDDLE  ALT=\"a+%2B+b+%2B+c=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ..........eq.1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6+-11+%2B+c=1+ ALIGN=MIDDLE  ALT=\"6+-11+%2B+c=1+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5+%2B+c=1+ ALIGN=MIDDLE  ALT=\"-5+%2B+c=1+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=1%2B5 ALIGN=MIDDLE  ALT=\"c=1%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28c=6%29 ALIGN=MIDDLE  ALT=\"highlight%28c=6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, your polynomial,or nth term formula is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+6n%5E2-11n%2B6 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+6n%5E2-11n%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" >  .......(for all terms given)\r<BR>\n" );
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document.write( "now use formula to find the next two numbers in this series:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B5%5D+=+6%2A5%5E2-11%2A5%2B6 ALIGN=MIDDLE  ALT=\"a%5B5%5D+=+6%2A5%5E2-11%2A5%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B5%5D+=+150-55%2B6 ALIGN=MIDDLE  ALT=\"a%5B5%5D+=+150-55%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B5%5D+=+101 ALIGN=MIDDLE  ALT=\"a%5B5%5D+=+101\"   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B6%5D+=+6%2A6%5E2-11%2A6%2B6 ALIGN=MIDDLE  ALT=\"a%5B6%5D+=+6%2A6%5E2-11%2A6%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B6%5D+=+216-66%2B6 ALIGN=MIDDLE  ALT=\"a%5B6%5D+=+216-66%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B6%5D+=+156 ALIGN=MIDDLE  ALT=\"a%5B6%5D+=+156\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "so, with these terms, your sequence is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27 ALIGN=MIDDLE  ALT=\"27\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=58 ALIGN=MIDDLE  ALT=\"58\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=101 ALIGN=MIDDLE  ALT=\"101\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=156 ALIGN=MIDDLE  ALT=\"156\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );