document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184006>Question 1184006</A>: <I><SPAN STYLE=\"word-break: break-all;\"> prove using mathematical induction:\r<BR>\n" );
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document.write( "1. 1^4 + 2^4 + 3^4 + ... + n^4 = (1/30)n(n+1)(2n+1)(3n^2 + 3n - 1)\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 #814585 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=814585\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I will go directly to step 3 of the Inductive process, and will assume the Inductive hypothesis to be true, i.e., \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E4+%2B+2%5E4+%2B+3%5E4+%2B+%22...%22+%2B+k%5E4+=+%281%2F30%29k%28k%2B1%29%282k%2B1%29%283k%5E2+%2B+3k+-+1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >  for some n = k.\r<BR>\n" );
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document.write( "Prove: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E4+%2B+2%5E4+%2B+3%5E4+%2B+%22...%22+%2B+k%5E4+=+%281%2F30%29k%28k%2B1%29%282k%2B1%29%283k%5E2+%2B+3k+-+1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > is true for n = k + 1, that is, \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E4+%2B+2%5E4+%2B+3%5E4+%2B+%22...%22+%2B+k%5E4+%2B+%28k%2B1%29%5E4+=+%281%2F30%29%28k%2B1%29%28k%2B2%29%282k%2B3%29%283%28k%2B1%29%5E2+%2B+3%28k%2B1%29+-+1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "From the inductive hypothesis, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E4+%2B+2%5E4+%2B+3%5E4+%2B+%22...%22+%2B+k%5E4+%2B+%28k%2B1%29%5E4++=+%281%2F30%29k%28k%2B1%29%282k%2B1%29%283k%5E2+%2B+3k+-+1%29+%2B+%28k%2B1%29%5E4+=+%28%28k%2B1%29%2F30%29%2Ared%28%28k%282k%2B1%29%283k%5E2+%2B+3k+-+1%29+%2B+30%28k%2B1%29%5E3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Focus on the the polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28p%28k%29%29+=+k%282k%2B1%29%283k%5E2+%2B+3k+-+1%29+%2B+30%28k%2B1%29%5E3 ALIGN=MIDDLE  ALT=\"red%28p%28k%29%29+=+k%282k%2B1%29%283k%5E2+%2B+3k+-+1%29+%2B+30%28k%2B1%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which is of 4th degree.\r<BR>\n" );
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document.write( "By the factor theorem, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%2B2 ALIGN=MIDDLE  ALT=\"k%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of this polynomial since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28-2%29+=+-2%2A%282%28-2%29%2B1%29%283%28-2%29%5E2+%2B+3%28-2%29+-+1%29+%2B+30%28%28-2%29%2B1%29%5E3+=+0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Again by the factor theorem, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2k%2B3 ALIGN=MIDDLE  ALT=\"2k%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of this polynomial since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28-3%2F2%29+=+%28-3%2F2%29%2A%282%28-3%2F2%29%2B1%29%283%28-3%2F2%29%5E2+%2B+3%28-3%2F2%29+-+1%29+%2B+30%28%28-3%2F2%29%2B1%29%5E3+=+0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28p%28k%29%29+=+red%28+%28k%2B2%29%282k%2B3%29p%5B2%5D%28k%29%29 ALIGN=MIDDLE  ALT=\"red%28p%28k%29%29+=+red%28+%28k%2B2%29%282k%2B3%29p%5B2%5D%28k%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5B2%5D%28k%29+=+ak%5E2%2Bbk+%2Bc ALIGN=MIDDLE  ALT=\"p%5B2%5D%28k%29+=+ak%5E2%2Bbk+%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a quadratic expression.\r<BR>\n" );
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document.write( "Now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28p%280%29%29+=+0%282%2A0%2B1%29%283%2A0%5E2+%2B+3%2A0+-+1%29+%2B+30%280%2B1%29%5E3+=red%28+%280%2B2%29%282%2A0%2B3%29%28a%2A0%5E2+%2B+b%2A0+%2Bc%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >   ===>   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30+=+6c ALIGN=MIDDLE  ALT=\"30+=+6c\"   DISABLED_style= \"max-width:90%; height:auto;\" >  ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=5 ALIGN=MIDDLE  ALT=\"c=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Also, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28k%29 ALIGN=MIDDLE  ALT=\"p%28k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a 4th degree polynimial whose leading term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6k%5E4 ALIGN=MIDDLE  ALT=\"6k%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Since  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29+=+%28k%2B2%29%282k%2B3%29%28ak%5E2%2Bbk+%2Bc%29 ALIGN=MIDDLE  ALT=\"p%28x%29+=+%28k%2B2%29%282k%2B3%29%28ak%5E2%2Bbk+%2Bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > also, this means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+3 ALIGN=MIDDLE  ALT=\"a+=+3\"   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=p%28k%29+=+k%282k%2B1%29%283k%5E2+%2B+3k+-+1%29+%2B+30%28k%2B1%29%5E3+=+red%28+%28k%2B2%29%282k%2B3%29%283k%5E2%2Bbk%2B5%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=p%281%29+=+1%2A3%283%2A1%5E2+%2B+3%2A1+-+1%29+%2B+30%281%2B1%29%5E3+=+red%28+%281%2B2%29%282%2A1%2B3%29%283%2A1%5E2%2Bb%2A1%2B5%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A5+%2B30%2A8+=+3%2A5%2A%28b%2B8%29 ALIGN=MIDDLE  ALT=\"3%2A5+%2B30%2A8+=+3%2A5%2A%28b%2B8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  ===>  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=9 ALIGN=MIDDLE  ALT=\"b=9\"   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=p%28k%29+=+red%28+%28k%2B2%29%282k%2B3%29%283k%5E2%2B9k%2B5%29%29+=+%28k%2B2%29%282k%2B3%29%283%28k%2B1%29%5E2%2B3%28k%2B1%29-1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Therefore,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E4+%2B+2%5E4+%2B+3%5E4+%2B+%22...%22+%2B+k%5E4+%2B+%28k%2B1%29%5E4++=+%281%2F30%29k%28k%2B1%29%282k%2B1%29%283k%5E2+%2B+3k+-+1%29+%2B+%28k%2B1%29%5E4+=+%28%28k%2B1%29%2F30%29%2Ared%28%28k%282k%2B1%29%283k%5E2+%2B+3k+-+1%29+%2B+30%28k%2B1%29%5E3%29%29+=+%28%28k%2B1%29%2F30%29%2A+%28k%2B2%29%282k%2B3%29%283%28k%2B1%29%5E2%2B3%28k%2B1%29-1%29++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "and the proof is complete.\r<BR>\n" );
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