Question 1093265
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1.nC1 + 2.nC2 + 3.nC3 + ........... +n.nCn


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<pre>
Let S be this sum:  S = {{{1*C[n]^1}}} + {{{2*C[n]^2}}} + {{{3*C[n]^3}}} + . . .  + {{{n*C[n]^n}}}.    (1)

Notice that S is the same as 

S = {{{0*C[n]^0}}} + {{{1*C[n]^1}}} + {{{2*C{n]^2}}} + {{{3*C[n]^3}}} + . . .  + {{{n*C[n]^n}}}.


Consider 2S and write it in this form 

2S = {{{0*C[n]^0}}} + {{{1*C[n]^1}}}       + {{{2*C[n]^2}}}        + {{{3*C[n]^3}}}         + . . .  + {{{n*C[n]^n}}} +

  +  {{{n*C[n]^n}}} + {{{(n-1)*C[n]^(n-1)}}} + {{{(n-2)*C[n]^(n-2)}}} + . . .   + {{{1*C[n]^1}}}     + {{{0*C[n]^0}}}.


Group the terms in pairs combining one term from the upper line and one term from the lower line. You will get

2S = {{{(0*C[n]^0 + n*C[n]^n) }}} + {{{(1*C[n]^1 + (n-1)*C[n]^(n-1)) }}} + {{{(2*C[n]^2 + (n-2)*C[n]^(n-2))}}} + {{{3*C[n]^3 + (n-3)*C[n]^(n-3))}}} + . . . + {{{(n*C[n]^n + 0*C[n]^0)}}}.     (2)


Use the fact that  {{{C[n]^k}}} = {{{C[n]^(n-k)}}}  for any k,  0 <= k <= n.

So you can rewrite (2) in the form

2S = {{{n*C[n]^0}}} + {{{n*C[n]^(n-1)}}} + {{{n*C[n]^2}}} + {{{n*C[n]^3}}} + . . . + {{{n*C[n]^n}}} = 

   = {{{n*(C[n]^0 + C[n]^1 + C[n]^2 + C[n]^3 + ellipsis + C[n]^n)}}}.


Now use well known fact that  

     {{{C[n]^0 + C[n]^1 + C[n]^2 + C[n]^3 + ellipsis + C[n]^n}}} = {{{2^n}}}.

     Regarding this fact, see the proof in the lesson <A HREF=https://www.algebra.com/algebra/homework/Permutations/Remarkable-identities-for-Binomial-Coefficients.lesson>Remarkable identities for Binomial Coefficients</A> in this site).


Then you finally get  

     2S = {{{n*2^n}}},   and, hence,  S = {{{n*2^(n-1)}}}.
</pre>

<U>Answer</U>.  &nbsp;&nbsp;{{{1*C[n]^1}}} + {{{2*C[n]^2}}} + {{{3*C[n]^3}}} + . . .  + {{{n*C[n]^n}}} = {{{n*2^(n-1)}}}.



 *** Solved ***



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On binomial coefficients see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Permutations/Binomial-Theorem.lesson>Binomial Theorem, Binomial Formula, Binomial Coefficients and Binomial Expansion</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Permutations/Remarkable-identities-for-Binomial-Coefficients.lesson>Remarkable identities for Binomial Coefficients</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Permutations/The-Pascal-triangle.lesson>The Pascal's triangle</A> 

in this site.


Also, you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this textbook under the topic
"<U>Binomial expansion, binomial coefficients, Pascal's triangle</U>".



Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.