document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=177933>Question 177933</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Which represents the common difference between successive terms in the sequence generated by tn=4/3n-6?<BR>\n" );
document.write( "Is it (a) -4/3 (b) 4/3 (c) -6 or (d)6<BR>\n" );
document.write( "I know I am over thinking this question :( </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #132931 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=132931\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The \"common difference\" is the difference between consecutive terms. So let's say we had the sequence: 1,2,3,4\r<BR>\n" );
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document.write( "The common difference is 1 since the difference between 1 and 2 is 1, the difference between 2 and 3 is 1, and the difference between 3 and 4 is 1. \r<BR>\n" );
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document.write( "So given any linear sequence, we can find the common difference by picking 2 terms and subtracting them. However, we need two terms from the sequence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5Bn%5D=%284%2F3%29n-6 ALIGN=MIDDLE  ALT=\"t%5Bn%5D=%284%2F3%29n-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's find the first term:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5Bn%5D=%284%2F3%29n-6 ALIGN=MIDDLE  ALT=\"t%5Bn%5D=%284%2F3%29n-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given sequence\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B0%5D=%284%2F3%29%280%29-6 ALIGN=MIDDLE  ALT=\"t%5B0%5D=%284%2F3%29%280%29-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=0 ALIGN=MIDDLE  ALT=\"n=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (a good number to start with). Note: you can start with any value of \"n\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B0%5D=0-6 ALIGN=MIDDLE  ALT=\"t%5B0%5D=0-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B0%5D=-6 ALIGN=MIDDLE  ALT=\"t%5B0%5D=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "So the first term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B0%5D=-6 ALIGN=MIDDLE  ALT=\"t%5B0%5D=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's find the second term:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5Bn%5D=%284%2F3%29n-6 ALIGN=MIDDLE  ALT=\"t%5Bn%5D=%284%2F3%29n-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given sequence\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B1%5D=%284%2F3%29%281%29-6 ALIGN=MIDDLE  ALT=\"t%5B1%5D=%284%2F3%29%281%29-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <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;\" > (the next term after <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B0%5D ALIGN=MIDDLE  ALT=\"t%5B0%5D\"   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=t%5B1%5D=4%2F3-6 ALIGN=MIDDLE  ALT=\"t%5B1%5D=4%2F3-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B1%5D=4%2F3-18%2F3 ALIGN=MIDDLE  ALT=\"t%5B1%5D=4%2F3-18%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 6 by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F3 ALIGN=MIDDLE  ALT=\"3%2F3\"   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=t%5B1%5D=%284-18%29%2F3 ALIGN=MIDDLE  ALT=\"t%5B1%5D=%284-18%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the fractions.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B1%5D=-14%2F3 ALIGN=MIDDLE  ALT=\"t%5B1%5D=-14%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "So the second term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B1%5D=-14%2F3 ALIGN=MIDDLE  ALT=\"t%5B1%5D=-14%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now take the difference of those two terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=t%5B1%5D-t%5B0%5D=-14%2F3-%28-6%29=-14%2F3%2B6=-14%2F3%2B18%2F3=%28-14%2B18%29%2F3=4%2F3 ALIGN=MIDDLE  ALT=\"d=t%5B1%5D-t%5B0%5D=-14%2F3-%28-6%29=-14%2F3%2B6=-14%2F3%2B18%2F3=%28-14%2B18%29%2F3=4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the common difference is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=4%2F3 ALIGN=MIDDLE  ALT=\"d=4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means that the answer is B)\r<BR>\n" );
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document.write( "Notice how the coefficient the term \"n\" is also <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F3 ALIGN=MIDDLE  ALT=\"4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This is no coincidence. It turns out that the coefficient in front of the term \"n\" is the common difference since you are adding multiples of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F3 ALIGN=MIDDLE  ALT=\"4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to -6.\r<BR>\n" );
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document.write( "So after all of that, we could have just said that the common difference is just the value in front of the \"n\". So once you understand how to find the common difference (and what it means), then you can quickly find the common difference.\r<BR>\n" );
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