document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205325>Question 1205325</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Simplify the following:\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 #842043 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=842043\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Tutor @ikleyn probably has the most efficient pathway, but here's another approach.\r<BR>\n" );
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document.write( "The pattern in the numerator (1^2,3^2,5^2,7^2,...) follows the sequence (2k-1)^2 where k is an integer and k = 1 is the starting index.<BR>\n" );
document.write( "Note how 1,3,5,7,... is an arithmetic sequence.<BR>\n" );
document.write( "The highest that k goes is k = 52 because 2k-1 = 2*52 - 1 = 103.\r<BR>\n" );
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document.write( "The pattern in the denominator (2,6,10,14,...) is also arithmetic and follows the sequence 4k-2<BR>\n" );
document.write( "Note that 4k-2 = 4*52 - 2 = 206.\r<BR>\n" );
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document.write( "Each term can be written of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%282k-1%29%5E2%29%2F%284k-2%29+=%28%282k-1%29%5E2%29%2F%282%282k-1%29%29+=+%282k-1%29%2F2+=+k+-+1%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We're tasked to find the summation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28%28k-1%2F2%29%2Ck=1%2C52%29 ALIGN=MIDDLE  ALT=\"sum%28%28k-1%2F2%29%2Ck=1%2C52%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Use these summation identities<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28%28A%2BB%29%2Ck=1%2Cn%29+=+%28sum%28A%2Ck=1%2Cn%29%29%2B%28sum%28B%2Ck=1%2Cn%29%29 ALIGN=MIDDLE  ALT=\"sum%28%28A%2BB%29%2Ck=1%2Cn%29+=+%28sum%28A%2Ck=1%2Cn%29%29%2B%28sum%28B%2Ck=1%2Cn%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28k%2Ck=1%2Cn%29+=+n%28n%2B1%29%2F2 ALIGN=MIDDLE  ALT=\"sum%28k%2Ck=1%2Cn%29+=+n%28n%2B1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%281%2C6%2Csum%28c%2Ck=1%2Cn%29+=+cn%2C%22where%22%2C%22c%22%2C%22is%22%2C%22a%22%2C%22constant%22%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "to have the following steps\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28%28k-1%2F2%29%2Ck=1%2C52%29=%28sum%28k%2Ck=1%2C52%29%29%2B%28sum%28-1%2F2%2Ck=1%2C52%29%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=sum%28%28k-1%2F2%29%2Ck=1%2C52%29=52%2A%2852%2B1%29%2F2%2B%28-1%2F2%29%2A52 ALIGN=MIDDLE  ALT=\"sum%28%28k-1%2F2%29%2Ck=1%2C52%29=52%2A%2852%2B1%29%2F2%2B%28-1%2F2%29%2A52\"   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=sum%28%28k-1%2F2%29%2Ck=1%2C52%29=1352 ALIGN=MIDDLE  ALT=\"sum%28%28k-1%2F2%29%2Ck=1%2C52%29=1352\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%281%2C5%2C%281%5E2%29%2F2%2B%283%5E2%29%2F6%2B%285%5E2%29%2F10%2B%287%5E2%29%2F14%2C%22%2B%22%2C%22...%22%2C%22%2B%22%2C%28103%5E2%29%2F206%29+=+1352 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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