document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1023504>Question 1023504</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove that<BR>\n" );
document.write( "1^2 + 3^2 + . . . + (2n &#8722; 1)^2 =1/3(4n^3 &#8722; n)<BR>\n" );
document.write( "for all natural numbers n. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #639029 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=639029\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E2+%2B+3%5E2+ ALIGN=MIDDLE  ALT=\"1%5E2+%2B+3%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >+ . . . + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282n+-+1%29%5E2+=%281%2F3%29%284n%5E3+-+n%29 ALIGN=MIDDLE  ALT=\"%282n+-+1%29%5E2+=%281%2F3%29%284n%5E3+-+n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Proof by induction.\r<BR>\n" );
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document.write( "For n = 1, the statement is trivially true.<BR>\n" );
document.write( "Assume the statement is true for some natural number n = k,<BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E2+%2B+3%5E2+ ALIGN=MIDDLE  ALT=\"1%5E2+%2B+3%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >+ . . . + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282k+-+1%29%5E2+=%281%2F3%29%284k%5E3+-+k%29 ALIGN=MIDDLE  ALT=\"%282k+-+1%29%5E2+=%281%2F3%29%284k%5E3+-+k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then by adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282k%2B1%29%5E2 ALIGN=MIDDLE  ALT=\"%282k%2B1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > on both sides, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E2+%2B+3%5E2+ ALIGN=MIDDLE  ALT=\"1%5E2+%2B+3%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >+ . . . + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282k+-+1%29%5E2+%2B+%282k%2B1%29%5E2=%281%2F3%29%284k%5E3+-+k%29%2B+%282k%2B1%29%5E2 ALIGN=MIDDLE  ALT=\"%282k+-+1%29%5E2+%2B+%282k%2B1%29%5E2=%281%2F3%29%284k%5E3+-+k%29%2B+%282k%2B1%29%5E2\"   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=%284k%5E3+-+k+%2B+3%282k%2B1%29%5E2%29%2F3+=+%284k%5E3-k%2B12k%5E2%2B12k%2B3%29%2F3+=+%28%284k%5E3%2B12k%5E2%2B12k%2B4%29+-+k+-1%29%2F3+=+%284%28k%2B1%29%5E3+-+%28k%2B1%29%29%2F3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, and the statement is proved...<BR>\n" );
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