Theres a quiz tomorrow and i have never felt soo lost.. please explain well every step.. thank you so much
Show that the following statemnet is true for all natural numbers n.
1 + 3 + 5 + ...+ (2n - 1 ) = n^2
I'll assume that you've ever seen this symbol: $\displaystyle \sum.$
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Start with the base case $\displaystyle n=1$ and prove that it's true. Now, let's assume our proposition for $\displaystyle n=k,$ hence, it follows that $\displaystyle \sum\limits_{i=1}^{k}{(2i-1)}=k^{2}.$ We need to prove that $\displaystyle \sum\limits_{i=1}^{k+1}{(2i-1)}=(k+1)^{2}.$ Here's the proof:
$\displaystyle \begin{aligned}
\sum\limits_{i=1}^{k+1}{(2i-1)}&=\sum\limits_{i=1}^{k}{(2i-1)}+\sum\limits_{i=k+1}^{k+1}{(2i-1)} \\
& =k^{2}+2k+1 \\
& =(k+1)^{2}.\quad\blacksquare
\end{aligned}$