document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=393642>Question 393642</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use mathematical induction to prove that<BR>\n" );
document.write( "1^3+2^3+3^3+....+n^3=(1+2+3+....+n)^2\r<BR>\n" );
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document.write( "Please I need your assistance </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #279525 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=279525\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The base case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E3+=+1%5E2 ALIGN=MIDDLE  ALT=\"1%5E3+=+1%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > satisfies. Now, suppose that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%5E3%2C+i+=+1%2C+n%29+=+%28sum%28i%2C+i+=+1%2C+n%29%29%5E2 ALIGN=MIDDLE  ALT=\"sum%28i%5E3%2C+i+=+1%2C+n%29+=+%28sum%28i%2C+i+=+1%2C+n%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for arbitrary n. We wish to prove that this implies the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2B1 ALIGN=MIDDLE  ALT=\"n%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > case will be true, or that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%5E3%2C+i+=+1%2C+n%2B1%29+=+%28sum%28i%2C+i+=+1%2C+n%2B1%29%29%5E2 ALIGN=MIDDLE  ALT=\"sum%28i%5E3%2C+i+=+1%2C+n%2B1%29+=+%28sum%28i%2C+i+=+1%2C+n%2B1%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We can show this is true, using the fact that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%5E3%2C+i+=+1%2C+n%2B1%29+=+sum%28%28i%5E3%29%2C+i+=+1%2C+n%29+%2B+%28n%2B1%29%5E3+=+%28sum%28i%2C+i+=+1%2C+n%29%29%5E2+%2B+%28n%2B1%29%5E3+=+%28%28n%28n%2B1%29%29%2F2%29%5E2+%2B+%28n%2B1%29%5E3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sum%28i%2C+i+=+1%2C+n%2B1%29%29%5E2+=+%28%28n%2B1%29%28n%2B2%29%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28sum%28i%2C+i+=+1%2C+n%2B1%29%29%5E2+=+%28%28n%2B1%29%28n%2B2%29%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Therefore this is equivalent to showing that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28n%28n%2B1%29%29%2F2%29%5E2+%2B+%28n%2B1%29%5E3+=+%28%28n%2B1%29%28n%2B2%29%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28%28n%28n%2B1%29%29%2F2%29%5E2+%2B+%28n%2B1%29%5E3+=+%28%28n%2B1%29%28n%2B2%29%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%2B1%29%5E2 ALIGN=MIDDLE  ALT=\"%28n%2B1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This holds for all n, so the statement holds for n+1. The induction is complete, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%5E3%2C+i+=+1%2C+n%29+=+%28sum%28i%2C+i+=+1%2C+n%29%29%5E2 ALIGN=MIDDLE  ALT=\"sum%28i%5E3%2C+i+=+1%2C+n%29+=+%28sum%28i%2C+i+=+1%2C+n%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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