document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1153676>Question 1153676</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Select the option for \"?\" that continues the pattern in each question.<BR>\n" );
document.write( "58<BR>\n" );
document.write( "3, -6, 12, 4, 20, ?<BR>\n" );
document.write( "13<BR>\n" );
document.write( "14<BR>\n" );
document.write( "6<BR>\n" );
document.write( "16 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #775940 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=775940\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "These numbers are related in the following way, beginning with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >: \r<BR>\n" );
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document.write( "First, we subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   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=3+-+9=+-6 ALIGN=MIDDLE  ALT=\"3+-+9=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " Next, we add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18 ALIGN=MIDDLE  ALT=\"18\"   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=-6+%2B+18=+12 ALIGN=MIDDLE  ALT=\"-6+%2B+18=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "  To get from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we subtract <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=12+-+8+=+4 ALIGN=MIDDLE  ALT=\"12+-+8+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and \r<BR>\n" );
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document.write( "to get from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the final number of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we must add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   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=4+%2B+16=+20 ALIGN=MIDDLE  ALT=\"4+%2B+16=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "  We need to examine the numbers that we add/take away in the sequence to find a pattern so we can predict the next number.\r<BR>\n" );
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document.write( "So, let's look at the relationship between the numbers we added and subtracted to get from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6+ ALIGN=MIDDLE  ALT=\"-6+\"   DISABLED_style= \"max-width:90%; height:auto;\" >to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and so on.  \r<BR>\n" );
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document.write( "First, we subtracted <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then added <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18 ALIGN=MIDDLE  ALT=\"18\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then subtracted <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then added <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Think of ways in which the numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18 ALIGN=MIDDLE  ALT=\"18\"   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;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" > are related.  \r<BR>\n" );
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document.write( "If I start with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, I have to multiply by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18 ALIGN=MIDDLE  ALT=\"18\"   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=9+%2A+2=+18 ALIGN=MIDDLE  ALT=\"9+%2A+2=+18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " To get from the next number, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to the fourth in the sequence, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" >, I must multiply by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > again: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8+%2A+2=16 ALIGN=MIDDLE  ALT=\"8+%2A+2=16\"   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
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document.write( "Now, we know the relationship between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18 ALIGN=MIDDLE  ALT=\"18\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the relationship between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  \r<BR>\n" );
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document.write( "We still need to figure out why the pattern moves from adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18 ALIGN=MIDDLE  ALT=\"18\"   DISABLED_style= \"max-width:90%; height:auto;\" > to subtracting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  To do this, I returned to my idea of multiplying by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which grouped our numbers as follows: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > was related to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18 ALIGN=MIDDLE  ALT=\"18\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > was related to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( " We can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9-1=8 ALIGN=MIDDLE  ALT=\"9-1=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  \r<BR>\n" );
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document.write( " So, let's say <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > is \"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\" The relationship between the numbers in the original sequence can be summed up as follows: \r<BR>\n" );
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document.write( "First number in the sequence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x ALIGN=MIDDLE  ALT=\"-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x ALIGN=MIDDLE  ALT=\"2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-+%28x-1%29 ALIGN=MIDDLE  ALT=\"-+%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2%28x-1%29 ALIGN=MIDDLE  ALT=\"+2%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  \r<BR>\n" );
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document.write( "Let's plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=first ALIGN=MIDDLE  ALT=\"first\"   DISABLED_style= \"max-width:90%; height:auto;\" > number in the sequence and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > to test our pattern: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+-+9=+-6 ALIGN=MIDDLE  ALT=\"3+-+9=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6+%2B+2%289%29=+-6+%2B+18=+12 ALIGN=MIDDLE  ALT=\"-6+%2B+2%289%29=+-6+%2B+18=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" >   <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12+-+%289-1%29=+12+-+8=+4 ALIGN=MIDDLE  ALT=\"12+-+%289-1%29=+12+-+8=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+4+%2B+2%289-1%29+=+4%2B+2%288%29+=+4+%2B+16=+20 ALIGN=MIDDLE  ALT=\"+4+%2B+2%289-1%29+=+4%2B+2%288%29+=+4+%2B+16=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "  If we put those answers in order, we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   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;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" >: the original sequence! \r<BR>\n" );
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document.write( "Using the pattern we found in that equation, we can predict that the next number in the sequence would be related to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" > by subtracting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29 ALIGN=MIDDLE  ALT=\"%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+ ALIGN=MIDDLE  ALT=\"x+\"   DISABLED_style= \"max-width:90%; height:auto;\" >still equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, since in order to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > after <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > we had to subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29 ALIGN=MIDDLE  ALT=\"%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20+-+%28x-2%29 ALIGN=MIDDLE  ALT=\"20+-+%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20+-+%289-2%29+=+20+-+7=+13 ALIGN=MIDDLE  ALT=\"20+-+%289-2%29+=+20+-+7=+13\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  \r<BR>\n" );
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document.write( "so, answer is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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