- #1
Robb
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- 8
Mod note: Fixed numerous problems with LaTeX
Also, the fractions shown throughout should instead be binomial coefficients
1. Homework Statement
prove the summation formula by counting a set in two ways
$$\sum_{k=0}^n k\left( \frac n k \right) = n *2^{n-1}$$
LHS = ##k \left(\frac n k \right) = k \left (\frac n 0 \right) + k \left( \frac n 1 \right) + k \left( \frac n 2 \right) + ...+ k \left(\frac n n \right) = k * 2^{n}##
RHS = ##n \left (\frac n k \right) * 2^{-1}##
Part two is just some analysis on my part. I'm not sure how to choose a set to count. A hint from the back of the book says to split the RHS into subsets corresponding to the term on the left. First attempt at latex so hope its not too confusing.
Also, the fractions shown throughout should instead be binomial coefficients
1. Homework Statement
prove the summation formula by counting a set in two ways
$$\sum_{k=0}^n k\left( \frac n k \right) = n *2^{n-1}$$
Homework Equations
LHS = ##k \left(\frac n k \right) = k \left (\frac n 0 \right) + k \left( \frac n 1 \right) + k \left( \frac n 2 \right) + ...+ k \left(\frac n n \right) = k * 2^{n}##
RHS = ##n \left (\frac n k \right) * 2^{-1}##
The Attempt at a Solution
Part two is just some analysis on my part. I'm not sure how to choose a set to count. A hint from the back of the book says to split the RHS into subsets corresponding to the term on the left. First attempt at latex so hope its not too confusing.
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