document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209827>Question 1209827</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a closed form for<BR>\n" );
document.write( "S_n = 1! \cdot (1^2 + 1) + 2! \cdot (2^2 + 2) + \dots + n! \cdot (n^2 + n).\]<BR>\n" );
document.write( "for any integer n \ge 1.  Your response should have a factorial. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850692 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850692\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's find a closed form for the sum S_n.\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Analyze the General Term**\r<BR>\n" );
document.write( "\n" );
document.write( "The general term of the sum is:\r<BR>\n" );
document.write( "\n" );
document.write( "T_k = k! * (k² + k)\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Simplify the General Term**\r<BR>\n" );
document.write( "\n" );
document.write( "We can factor out k from the parentheses:\r<BR>\n" );
document.write( "\n" );
document.write( "T_k = k! * k(k + 1)\r<BR>\n" );
document.write( "\n" );
document.write( "Rearrange the terms:\r<BR>\n" );
document.write( "\n" );
document.write( "T_k = k * (k + 1) * k!\r<BR>\n" );
document.write( "\n" );
document.write( "Notice that (k + 1) * k! = (k + 1)!\r<BR>\n" );
document.write( "\n" );
document.write( "So, T_k = k * (k + 1)!\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Rewrite k**\r<BR>\n" );
document.write( "\n" );
document.write( "We can rewrite k as (k + 2 - 2):\r<BR>\n" );
document.write( "\n" );
document.write( "T_k = (k + 2 - 2) * (k + 1)!\r<BR>\n" );
document.write( "\n" );
document.write( "Distribute:\r<BR>\n" );
document.write( "\n" );
document.write( "T_k = (k + 2) * (k + 1)! - 2 * (k + 1)!\r<BR>\n" );
document.write( "\n" );
document.write( "Recognize that (k + 2) * (k + 1)! = (k + 2)!\r<BR>\n" );
document.write( "\n" );
document.write( "So, T_k = (k + 2)! - 2 * (k + 1)!\r<BR>\n" );
document.write( "\n" );
document.write( "**4. Apply the Summation**\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = Σ[k=1 to n] T_k\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = Σ[k=1 to n] [(k + 2)! - 2 * (k + 1)!]\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = [3! - 2 * 2!] + [4! - 2 * 3!] + [5! - 2 * 4!] + ... + [(n + 2)! - 2 * (n + 1)!]\r<BR>\n" );
document.write( "\n" );
document.write( "**5. Observe the Telescoping Pattern**\r<BR>\n" );
document.write( "\n" );
document.write( "Notice that many terms cancel out:\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = [3! - 2 * 2!] + [4! - 2 * 3!] + [5! - 2 * 4!] + ... + [(n + 2)! - 2 * (n + 1)!]\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = -2 * 2! + (3! - 2 * 3!) + (4! - 2 * 4!) + ... + (n + 2)!\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = -2 * 2! -3! -4! ... +(n+2)!\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = (n + 2)! - (2(2!) + 3! + 4! + ... + (n+1)!)\r<BR>\n" );
document.write( "\n" );
document.write( "Now let's examine a different approach.\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = Σ[k=1 to n] k * (k + 1)!\r<BR>\n" );
document.write( "\n" );
document.write( "We can rewrite k as (k + 1 - 1):\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = Σ[k=1 to n] (k + 1 - 1) * (k + 1)!\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = Σ[k=1 to n] [(k + 1) * (k + 1)! - (k + 1)!]\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = Σ[k=1 to n] [(k + 2)! - (k + 1)!]\r<BR>\n" );
document.write( "\n" );
document.write( "This is a telescoping sum:\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = [3! - 2!] + [4! - 3!] + [5! - 4!] + ... + [(n + 2)! - (n + 1)!]\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = (n + 2)! - 2!\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = (n + 2)! - 2\r<BR>\n" );
document.write( "\n" );
document.write( "**Final Answer:**\r<BR>\n" );
document.write( "\n" );
document.write( "S_n = (n + 2)! - 2<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );