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\title{\bf Assignment 1 \\[2ex]
\rm\normalsize CS 311, Fall 2015 \\ Due October 5, 2015}
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\paragraph{Problem 1} Prove by induction on $n$:
\[ \forall a \geq 2, \sum_{i=0}^n a^i = \frac{a^{n+1} -1}{a-1}\]

\paragraph{Problem 2} Use the {\it pigeonhole principle} to prove the following: Every undirected 
    simple graph with at least 2 nodes has two nodes with the same degree.
    
\paragraph{Problem 3} Let $S = \{ a,b,c,d,e,f \}$, find the number of subsets of $S$ which 
    include the elements $a$ or $b$.

\paragraph{Problem 4} Given two natural numbers $x$ and $y$ such that $x > y$, prove that:
\[ x \bmod{y} < \frac{x}2\]

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