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Thread: Mathematical Induction

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    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
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by NeedHelp18 View Post
    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)
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    I'll assume that you've ever seen this symbol: $\displaystyle \sum.$

    -----

    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:

    $\displaystyle \begin{aligned}
    \sum\limits_{i=1}^{k+1}{(2i-1)}&=\sum\limits_{i=1}^{k}{(2i-1)}+\sum\limits_{i=k+1}^{k+1}{(2i-1)} \\
    & =k^{2}+2k+1 \\
    & =(k+1)^{2}.\quad\blacksquare
    \end{aligned}$
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