1. ## Mathematical Induction

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

2. Originally Posted by NeedHelp18
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
see problem 1 in the attachment (I found it using google by the way. this is a standard problem)

3. 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}