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{\bf Name: }
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{\bf Section: }

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\centerline {\bf Computer Science 243}
\centerline {\bf Spring 2025}
\centerline {\bf Homework 8}
% \centerline{\bf My name}
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\bf Due: 11:59 p.m., Wednesday, 4/2/25 \hfill Points: 50
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Answer the following questions and show your work.  Your final submission must be completely your own work.

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\item {[10 points]} Prove the following by mathematical induction whenever $n$ is a positive integer:
\[ \sum_{i=1}^{n} (4i + 2) = 2n(n + 2) \]


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\item {[10 points]} Prove by induction that ${n^3} < n!$ for all integers $n \ge 6$. 
\textit {Hint: Use Pascal's triangle where needed.}\\


\item {[10 points]} Prove the following by mathematical induction whenever $n$ is a positive integer:
\[ \sum_{i=1}^{n} {(6i)}^2 = 6n(n + 1)(2n + 1) \]


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\item {[6 points]}  Give a recursive definition of the following:
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\item the set $S$ of positive integers that are multiples of 4
\item the set $S$ of positive integer powers of 5
\item the set $S$ of positive integers congruent to 6 modulo 11
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\item {[14 points]}  For the sequence defined by $a_n= n(n-3)$ for $n \ge 3$ :  
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\item {[4 points]} give a recursive definition of the sequence
\item {[10 points]} use an inductive proof to show that the recursive definition in (a) is equivalent to the original closed-form expression
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