document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=569522>Question 569522</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If there are (2n+1) terms in an arithmetic series, prove that the ratio of the sum of odd place terms to the sum of even place terms is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+n%5E4-1+ ALIGN=MIDDLE  ALT=\"+n%5E4-1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+n%5E4-n%5E3%2Bn%5E2-n+ ALIGN=MIDDLE  ALT=\"+n%5E4-n%5E3%2Bn%5E2-n+\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #367510 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=367510\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Suppose our sequence is a, a+d, a+2d, ..., a+2nd (this ensures that there are 2n+1 terms). Denote S_o and S_e to be the sum of the odd place terms and even place terms respectively. Then,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE S_o = a+(a+2d) + (a+4d) + ... + (a+2nd)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE S_e = (a+d) + (a+3d) + ... + (a+(2n-1)d)\">\r<BR>\n" );
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document.write( "S_o includes n+1 terms and S_e includes n terms. By equating the coefficients of a and d in each series, we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE S_o = a(n+1) + n(n+1)d\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE S_e = an + n^2d\">\r<BR>\n" );
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document.write( "Therefore, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{S_o}{S_e} = \frac{a(n+1) + n(n+1)d}{an + n^2d} = \frac{(n+1)(a+nd)}{n(a+nd)} = \frac{n+1}{n}\">. Multiplying both numerator and denominator by n^3 - n^2 + n - 1 yields the desired result. </SPAN>\n" );
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