induction_quiz.tex

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\activitytitle{Quiz on induction}{15 points}

\noindent
For each problem below, clearly state $P(1), P(k),$ and $P(k+1)$ as logical statements with double quotes around them.
When proving that $P(k)$ being true implies that $P(k+1)$ is true, do not write down $P(k+1)$ as if it were true, but rather start with one side and work with it until it turns into the other side.

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\showNN{Use induction to show that for $n > 0$, 8 divides $5^n + 2(3^{n-1}) + 1.$
{\bf Hint:}  As in other proofs of divisibility, add and subtract to be able to use $P(n)$ to simplify $P(n+1)$.}{6in}



\showNN{On the back of this piece of paper, use induction to show that for all $n \geq 1$, we have that $1(1!) + 2(2!) + \cdots + n(n!) = (n+1)! - 1$.}{0in}

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