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August 29th, 2017, 10:34 PM  #1 
Senior Member Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0  Show that the sum is equal to the value I'm not quite sure where to go with this one. The problem hints at using the binomial expansion, but I'm struggling to see how it will help me here. I tried writing out a couple of terms to no avail as well as expanded the combinatoric portion. I would greatly appreciate any strategies you may have to offer. 
August 29th, 2017, 11:05 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 21,127 Thanks: 2336 
Write the binomial expansion of $(1 + x)^n$, then list the differences between that polynomial and the lefthand side of the equation you are given to prove.

August 29th, 2017, 11:48 PM  #3  
Senior Member Joined: Sep 2015 From: USA Posts: 2,647 Thanks: 1476  Quote:
differentiate w/respect to $x$ $n(1+x)^{n1} = \displaystyle \sum \limits_{k=1}^{n}k \dbinom{n}{k} x^{k1}$ let $x=1$ $n2^{n1} = \displaystyle \sum \limits_{k=1}^{n} k \dbinom{n}{k}$  

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