Math 302 HW 0 - B. Rodriguez
Auteur:
Breanna Rodriguez
Last Updated:
il y a 5 ans
License:
Creative Commons CC BY 4.0
Résumé:
HW 0
\begin
Discover why 18 million people worldwide trust Overleaf with their work.
HW 0
\begin
Discover why 18 million people worldwide trust Overleaf with their work.
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\begin{document}
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\noindent
\textbf{MATH 302-01} \hfill \textbf{Breanna Rodriguez} \\
\normalsize Prof. Soto \hfill Due Date: 08/25/19 \\
% This is where you name your homework
\begin{center}
\textbf{Homework 0}
\end{center}
% This is how you call the environment for the statement to be proved.
\begin{statement}{\#1}
Prove that
\begin{equation}
\label{bree}
1 + 2 + 3+...+ n = \sum_{k=1}^n k = \frac{n(n+1)}{2}.
\end{equation}
\end{statement}
% This is how you call the proof environment
\begin{proof}
We would like to prove that $ \sum_{k=1}^n k = \frac{n(n+1}{2}$ so we proceed by induction. \\
\textbf{Base Case}. For $n = 1$ we see that the left hand side of (\ref{bree}) is 1 whereas the right hand side is given by
$$ \frac{1(1+1)}{2}= 1. $$
as well. Hence the statement is true for $n =1$ \\
\textbf{Induction Case}. Assume that the statement holds for n+1, that is we need to show that
\begin{align*}
1 + 2 + 3 + \dotsc + n + (n+1) &= \frac{(n+1)((n+1)+1)}{2} \\
& = \frac{(n+1)(n+2)}{2}
\end{align*}
Thus we will begin with the left hand side of (\ref{bree}) to reach our conclusion. By our assumption we know that
$$ 1 + 2 + 3 + \dotsc + n + (n+1) = \frac{n(n+1)}{2} + (n+1)
$$
Thus we use a bit of algebra as follows to reach our conclusion:
\begin{align*}
1 + 2 + 3 + \dotsc + n+ (n+1) & = \frac{n(n+1)}{2}+ (n+1) \\
& = \frac{n(n+1)}{2} + \frac{2(n+1)}{2} \\
& = \frac{(n+1)(n+2)}{2}.
\end{align*}
Thus by induction we see that statement (\ref{bree}) is true.
\end{proof}
\end{document}