document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209826>Question 1209826</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For a positive integer k, let<BR>\n" );
document.write( "S_k = 1 \cdot 1! \cdot 2 + 2 \cdot 2! \cdot 3 + \dots + k \cdot k! \cdot (k + 1).<BR>\n" );
document.write( "Find a closed form for S_k. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851950 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851950\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "S_k = 1*(1^2) + 2!*(2^2)*3 + . . .  + k*k!*(k + 1).<BR>\n" );
document.write( "Find a closed form for S_k.<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The solution and the answer in the post by &nbsp;@CPhill both are &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28INCORRECT%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28INCORRECT%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Indeed, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;let's check for &nbsp;k = 3.\r<BR>\n" );
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document.write( "Left side is\r<BR>\n" );
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document.write( "  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1!*(1^2) + 2!*(2^2)*3 + 3*3!*(3+1) = 1*(1) + 2*(4)*3 + 3*(6)*4 = 1 + 24 + 72 = 97.\r<BR>\n" );
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document.write( "Right side, according to @CPhill, is \r<BR>\n" );
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document.write( "  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3*(3+1)! - 2 = 3*4! - 2 = 3*24 - 2 = 72 - 2 = 70.\r<BR>\n" );
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document.write( "But &nbsp;&nbsp;97 =/= 70.\r<BR>\n" );
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document.write( "This is the &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28CONTRADICTION%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28CONTRADICTION%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;which ruins the solution by &nbsp;@CPhill to dust.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Regarding the post by @CPhill . . . \r<BR>\n" );
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document.write( "Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.\r<BR>\n" );
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document.write( "The artificial intelligence is like a baby now. It is in the experimental stage <BR>\n" );
document.write( "of development and can make mistakes and produce nonsense without any embarrassment.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It has no feeling of shame - it is shameless.\r<BR>\n" );
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document.write( "This time, again, &nbsp;it made an error.\r<BR>\n" );
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document.write( "Although the @CPhill' solutions are copy-paste &nbsp;Google &nbsp;AI solutions, &nbsp;there is one essential difference.\r<BR>\n" );
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document.write( "Every time, &nbsp;Google &nbsp;AI &nbsp;makes a note at the end of its solutions that &nbsp;Google &nbsp;AI &nbsp;is experimental<BR>\n" );
document.write( "and can make errors/mistakes.\r<BR>\n" );
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document.write( "All @CPhill' solutions are copy-paste of &nbsp;Google &nbsp;AI &nbsp;solutions, with one difference:<BR>\n" );
document.write( "@PChill never makes this notice and never says that his solutions are copy-past that of Google.<BR>\n" );
document.write( "So, &nbsp;he &nbsp;NEVER &nbsp;SAYS &nbsp;TRUTH.\r<BR>\n" );
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document.write( "Every time, &nbsp;@CPhill embarrassed to tell the truth.\r<BR>\n" );
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document.write( "But I am not embarrassing to tell the truth, &nbsp;as it is my duty at this forum.\r<BR>\n" );
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document.write( "And the last my comment.\r<BR>\n" );
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document.write( "When you obtain such posts from @CPhill, &nbsp;remember, &nbsp;that &nbsp;NOBODY &nbsp;is responsible for their correctness, <BR>\n" );
document.write( "until the specialists and experts will check and confirm their correctness.\r<BR>\n" );
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document.write( "Without it, &nbsp;their reliability is &nbsp;ZERO and their creadability is &nbsp;ZERO, &nbsp;too.\r<BR>\n" );
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