document.write( "Question 1093265: Evaluate \r
\n" ); document.write( "\n" ); document.write( "1.nC1+2.nC2+3.nC3+...........+n.nCn \n" ); document.write( "

Algebra.Com's Answer #707906 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "1.nC1 + 2.nC2 + 3.nC3 + ........... +n.nCn\r
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document.write( "Let S be this sum:  S =  +  +  + . . .  + .    (1)\r\n" );
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document.write( "Notice that S is the same as \r\n" );
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document.write( "S =  +  +  +  + . . .  + .\r\n" );
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document.write( "Consider 2S and write it in this form \r\n" );
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document.write( "2S =  +        +         +          + . . .  +  +\r\n" );
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document.write( "  +   +  +  + . . .   +      + .\r\n" );
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document.write( "Group the terms in pairs combining one term from the upper line and one term from the lower line. You will get\r\n" );
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document.write( "2S =  +  +  +  + . . . + .     (2)\r\n" );
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document.write( "Use the fact that   =   for any k,  0 <= k <= n.\r\n" );
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document.write( "So you can rewrite (2) in the form\r\n" );
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document.write( "2S =  +  +  +  + . . . +  = \r\n" );
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document.write( "   = .\r\n" );
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document.write( "Now use well known fact that  \r\n" );
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document.write( "      = .\r\n" );
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document.write( "     Regarding this fact, see the proof in the lesson Remarkable identities for Binomial Coefficients in this site).\r\n" );
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document.write( "Then you finally get  \r\n" );
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document.write( "     2S = ,   and, hence,  S = .\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answer.    $\"1%2AC%5Bn%5D%5E1\"$ + $\"2%2AC%5Bn%5D%5E2\"$ + $\"3%2AC%5Bn%5D%5E3\"$ + . . . + $\"n%2AC%5Bn%5D%5En\"$ = $\"n%2A2%5E%28n-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( " *** Solved ***\r
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\n" ); document.write( "On binomial coefficients see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Binomial Theorem, Binomial Formula, Binomial Coefficients and Binomial Expansion\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this textbook under the topic
\n" ); document.write( "\"Binomial expansion, binomial coefficients, Pascal's triangle\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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