document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1085377>Question 1085377</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1) Find the 11th term of this sequence. -10,20,-40,80,...<BR>\n" );
document.write( "2) Find the next two terms of the following sequence: 14,38,74,122,182,254,...<BR>\n" );
document.write( "3) Find the next two terms of the following sequence: -21,-9,11,39,...<BR>\n" );
document.write( "4) What is the 9th term in the geometric sequence in which a3 is 36 and a6 is 972?<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 #699428 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=699428\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">I'll do the first two problems to get you started. In the future, please post one problem at a time.\r<BR>\n" );
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document.write( "Problem 1)\r<BR>\n" );
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document.write( "This is a geometric sequence since the ratio of terms are all equal to -2\r<BR>\n" );
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document.write( "(term 2)/(term 1) = (20)/(-10) = -2<BR>\n" );
document.write( "(term 3)/(term 2) = (-40)/(20) = -2<BR>\n" );
document.write( "(term 4)/(term 3) = (80)/(-40) = -2\r<BR>\n" );
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document.write( "We could keep multiplying each term by -2 to generate enough terms to get to term 11, but that is a bit more work than needed.\r<BR>\n" );
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document.write( "Instead, let's form the nth term formula to get \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+a%2A%28r%29%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+a%2A%28r%29%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+-10%2A%28-2%29%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+-10%2A%28-2%29%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note how <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+-10 ALIGN=MIDDLE  ALT=\"a+=+-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the first term and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+-2 ALIGN=MIDDLE  ALT=\"r+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the common ratio previously found.\r<BR>\n" );
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document.write( "Now plug <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+11 ALIGN=MIDDLE  ALT=\"n+=+11\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the nth term formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+-10%2A%28-2%29%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+-10%2A%28-2%29%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B11%5D+=+-10%2A%28-2%29%5E%2811-1%29 ALIGN=MIDDLE  ALT=\"a%5B11%5D+=+-10%2A%28-2%29%5E%2811-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B11%5D+=+-10%2A%28-2%29%5E%2810%29 ALIGN=MIDDLE  ALT=\"a%5B11%5D+=+-10%2A%28-2%29%5E%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B11%5D+=+-10%2A%281024%29 ALIGN=MIDDLE  ALT=\"a%5B11%5D+=+-10%2A%281024%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B11%5D+=+-10240 ALIGN=MIDDLE  ALT=\"a%5B11%5D+=+-10240\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The 11th term is <font color=red>-10240</font>\r<BR>\n" );
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document.write( "Problem 2)\r<BR>\n" );
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document.write( "At first glance, this sequence doesn't seem to have a pattern. It's certainly not arithmetic since\r<BR>\n" );
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document.write( "(term 2) - (term 1) = 38 - 14 = 24<BR>\n" );
document.write( "(term 3) - (term 2) = 74 - 38 = 36\r<BR>\n" );
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document.write( "the differences aren't the same meaning we don't have a common difference.\r<BR>\n" );
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document.write( "It's also not geometric because\r<BR>\n" );
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document.write( "(term 2)/(term 1) = 38/14 = 2.71 (approx)<BR>\n" );
document.write( "(term 3)/(term 2) = 74/38 = 1.95 (approx)\r<BR>\n" );
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document.write( "meaning that there isn't a common ratio either. \r<BR>\n" );
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document.write( "It turns out that this sequence is following a <a  rel=nofollow HREF=\"http://wtmaths.com/quadratic_progressions.html\">quadratic progression</a>. The first level of differences (differences between adjacent terms) are\r<BR>\n" );
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document.write( "24, 36, 48, 60, 72\r<BR>\n" );
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document.write( "which are the results of subtracting each previous term off from its next term. Now focus on the sequence of differences. This sequence {24, 36, 48, 60, 72, ...} is arithmetic with common difference d = 12. This second level difference implies that we have a quadratic progression.\r<BR>\n" );
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document.write( "I'm skipping a few steps but we can use technology to construct the equation to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+6x%5E2+%2B+18x+%2B+14 ALIGN=MIDDLE  ALT=\"y+=+6x%5E2+%2B+18x+%2B+14\"   DISABLED_style= \"max-width:90%; height:auto;\" > where x is the term number and y is the term itself. \r<BR>\n" );
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document.write( "Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+7 ALIGN=MIDDLE  ALT=\"x+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+6x%5E2+%2B+18x+%2B+14 ALIGN=MIDDLE  ALT=\"y+=+6x%5E2+%2B+18x+%2B+14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+6%287%29%5E2+%2B+18%287%29+%2B+14 ALIGN=MIDDLE  ALT=\"y+=+6%287%29%5E2+%2B+18%287%29+%2B+14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+434 ALIGN=MIDDLE  ALT=\"y+=+434\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Repeat for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+8 ALIGN=MIDDLE  ALT=\"x+=+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=y+=+6x%5E2+%2B+18x+%2B+14 ALIGN=MIDDLE  ALT=\"y+=+6x%5E2+%2B+18x+%2B+14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+6%288%29%5E2+%2B+18%288%29+%2B+14 ALIGN=MIDDLE  ALT=\"y+=+6%288%29%5E2+%2B+18%288%29+%2B+14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+542 ALIGN=MIDDLE  ALT=\"y+=+542\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Therefore the next two terms are <font color=red>434 and 542</font></font> </SPAN>\n" );
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