document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1036497>Question 1036497</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Alpha writes the infinite arithmetic sequence<BR>\n" );
document.write( "10, 8, 6, 4, 2, 0 ...<BR>\n" );
document.write( "Beta writes the infinite geometric sequence<BR>\n" );
document.write( "9, 6, 4, 8/3, 16/9 ...<BR>\n" );
document.write( "Gamma makes a sequence whose nth term is the product of the nth term of Alpha's sequence and the nth term of Beta's sequence:<BR>\n" );
document.write( "10 * 9, 8 * 6, 6 * 4, 4 * 8/3, 2 * 16/9, ...<BR>\n" );
document.write( "What is the sum of Gamma's entire sequence? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #651235 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=651235\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The alpha sequence is an arithmetic sequence with first term 10 and common difference -2, and is defined by the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+12-2n+=+2%286-n%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+12-2n+=+2%286-n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for n = 1,2,3,... .\r<BR>\n" );
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document.write( "The beta sequence is a geometric sequence with first term 9 and common ratio 2/3, and is defined by the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Bn%5D+=+9%2A%282%2F3%29%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"b%5Bn%5D+=+9%2A%282%2F3%29%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The gamma sequence is thus defined by the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%5Bn%5D+=+2%286-n%29%2A+9%2A%282%2F3%29%5E%28n-1%29+=+18%286-n%29%2A%282%2F3%29%5E%28n-1%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "We have to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%2818%286-n%29%2A%282%2F3%29%5E%28n-1%29%2Cn+=+1%2C+infinity%29 ALIGN=MIDDLE  ALT=\"sum%2818%286-n%29%2A%282%2F3%29%5E%28n-1%29%2Cn+=+1%2C+infinity%29\"   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=18sum%28%286%2A%282%2F3%29%5E%28n-1%29-n%2A%282%2F3%29%5E%28n-1%29%29%2Cn+=+1%2C+infinity%29 ALIGN=MIDDLE  ALT=\"18sum%28%286%2A%282%2F3%29%5E%28n-1%29-n%2A%282%2F3%29%5E%28n-1%29%29%2Cn+=+1%2C+infinity%29\"   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=18sum%286%2A%282%2F3%29%5E%28n-1%29%2Cn+=+1%2C+infinity%29-18sum%28n%2A%282%2F3%29%5E%28n-1%29%2C+n=1%2C+infinity%29 ALIGN=MIDDLE   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=108sum%28%282%2F3%29%5E%28n-1%29%2Cn+=+1%2C+infinity%29-18sum%28n%2A%282%2F3%29%5E%28n-1%29%2C+n=1%2C+infinity%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The infinite geometric series <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Bx%2Bx%5E2%2Bx%5E3 ALIGN=MIDDLE  ALT=\"1%2Bx%2Bx%5E2%2Bx%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >+... = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%281-x%29 ALIGN=MIDDLE  ALT=\"1%2F%281-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as long as -1 < x < 1.<BR>\n" );
document.write( "Staying within the domain of convergence and differentiating term-by-term, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B2x%2B3x%5E2%2B4x%5E3 ALIGN=MIDDLE  ALT=\"1%2B2x%2B3x%5E2%2B4x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >+... = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%281-x%29%5E2 ALIGN=MIDDLE  ALT=\"1%2F%281-x%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now the first infinite sum is an infinite geometric series with sum <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%281-2%2F3%29+=+1%2F%281%2F3%29+=+3 ALIGN=MIDDLE  ALT=\"1%2F%281-2%2F3%29+=+1%2F%281%2F3%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The second infinite sum is derivative of the infinite geometric series and has sum <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%281-2%2F3%29%5E2+=+1%2F%281%2F3%29%5E2+=+1%2F%281%2F9%29+=+9 ALIGN=MIDDLE  ALT=\"1%2F%281-2%2F3%29%5E2+=+1%2F%281%2F3%29%5E2+=+1%2F%281%2F9%29+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Therefore...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=108sum%28%282%2F3%29%5E%28n-1%29%2Cn+=+1%2C+infinity%29-18sum%28n%2A%282%2F3%29%5E%28n-1%29%2C+n=1%2C+infinity%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= 108*3 - 18*9 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28162%29 ALIGN=MIDDLE  ALT=\"highlight%28162%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " </SPAN>\n" );
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