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Name (Last, First): $\underline{\hspace{3in}}$ \hfill Student ID: $\underline{\hspace{2in}}$

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{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1415}{2014-15}:
  \href{\myurl/classes/15-475-ProblemSolving}{MAT 475 Problem Solving Seminar}:
  \hfill\url{http://drorbn.net/15-475}
}

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{\large\bf Quiz 11} on April 2, 2015: ``Generalize''. You have 25 minutes to solve the two problems below. Please write on both sides of the page. \hfill {\bf Good Luck!}

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\noindent{\bf Problem 1.} Compute the sums
\hfill(a)\qquad$\displaystyle\sum_{k=1}^n k(k-1)\binom{n}{k}$
\hfill and\hfill(b)\qquad$\displaystyle\sum_{k=1}^n \frac{1}{k+1}\binom{n}{k}$.\hfill\null

\vskip 2mm\noindent{\bf Problem 2} (Larson's 2.4.3, modified). Let $F_n$ denote the Fibonacci numbers, defined by
$F_0=F_1=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 2$. Prove that $F_{2n}=(F_n)^2+(F_{n-1})^2$.

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