document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1104186>Question 1104186</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the sum of  1/2(4) +1/3(5) + 1/4(6) + ..... + 1/13(15)\r<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 #718957 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=718957\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Each of the fractions is of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28n%28n%2B2%29%29 ALIGN=MIDDLE  ALT=\"1%2F%28n%28n%2B2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Whenever you see a problem like this, where you are adding several terms of the same form, it is almost certain that you can write each term in such a way that, when all the terms are added, most of the terms cancel, leaving you with a relatively simple answer.<br><BR>\n" );
document.write( "So let's find a way to \"decompose\" the general term into the sum of two terms.  Since the denominator is n(n+2), we want to write the term as the sum of two fractions, one with denominator n and the other with denominator (n+2).<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28n%28n%2B2%29%29+=+A%2Fn+%2B+B%2F%28n%2B2%29 ALIGN=MIDDLE  ALT=\"1%2F%28n%28n%2B2%29%29+=+A%2Fn+%2B+B%2F%28n%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fn%28n%2B2%29+=+%28A%28n%2B2%29+%2B+B%28n%29%29%2F%28n%28n%2B2%29%29 ALIGN=MIDDLE  ALT=\"1%2Fn%28n%2B2%29+=+%28A%28n%2B2%29+%2B+B%28n%29%29%2F%28n%28n%2B2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fn%28n%2B2%29+=+%28%28A%2BB%29n+%2B+2A%29%2F%28n%28n%2B2%29%29 ALIGN=MIDDLE  ALT=\"1%2Fn%28n%2B2%29+=+%28%28A%2BB%29n+%2B+2A%29%2F%28n%28n%2B2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+%28A%2BB%29n+%2B+2A ALIGN=MIDDLE  ALT=\"1+=+%28A%2BB%29n+%2B+2A\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This equation must be true for all values of n; that means<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2BB+=+0 ALIGN=MIDDLE  ALT=\"A%2BB+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2A+=+1 ALIGN=MIDDLE  ALT=\"2A+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Solving that system of equations gives us<BR>\n" );
document.write( "A = 1/2; B = -1/2<br><BR>\n" );
document.write( "So the general term of the sequence can be written as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28n%28n%2B2%29%29+=+%281%2F2%29%2Fn+-+%281%2F2%29%2F%28n%2B2%29+=+%281%2F2%29%281%2Fn-1%2F%28n%2B2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now rewrite each term as the sum of two terms, using this pattern:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2%284%29+=+%281%2F2%29%281%2F2-1%2F4%29 ALIGN=MIDDLE  ALT=\"1%2F2%284%29+=+%281%2F2%29%281%2F2-1%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3%285%29+=+%281%2F2%29%281%2F3-1%2F5%29 ALIGN=MIDDLE  ALT=\"1%2F3%285%29+=+%281%2F2%29%281%2F3-1%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4%286%29+=+%281%2F2%29%281%2F4-1%2F6%29 ALIGN=MIDDLE  ALT=\"1%2F4%286%29+=+%281%2F2%29%281%2F4-1%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F5%287%29+=+%281%2F2%29%281%2F5-1%2F7%29 ALIGN=MIDDLE  ALT=\"1%2F5%287%29+=+%281%2F2%29%281%2F5-1%2F7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F11%2813%29+=+%281%2F2%29%281%2F11-1%2F13%29 ALIGN=MIDDLE  ALT=\"1%2F11%2813%29+=+%281%2F2%29%281%2F11-1%2F13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F12%2814%29+=+%281%2F2%29%281%2F12-1%2F14%29 ALIGN=MIDDLE  ALT=\"1%2F12%2814%29+=+%281%2F2%29%281%2F12-1%2F14%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F13%2815%29+=+%281%2F2%29%281%2F13-1%2F15%29 ALIGN=MIDDLE  ALT=\"1%2F13%2815%29+=+%281%2F2%29%281%2F13-1%2F15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now looking at the terms in parentheses, you see that there is a 1/4 and a -1/4, a 1/5 and a -1/5, ..., and a 1/13 and a -1/13.<br><BR>\n" );
document.write( "So all those terms cancel; all that is left is the first fractions from the original first two terms and the second fractions from the original last two terms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%281%2F2%2B1%2F3-1%2F14-1%2F15%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29%281%2F2%2B1%2F3-1%2F14-1%2F15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So, using a common denominator, the sum we are looking for is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28105%2F210%2B70%2F210-15%2F210-14%2F210%29+=+%281%2F2%29%28146%2F210%29+=+73%2F210 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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