# Mathematical Induction

• Oct 26th 2008, 12:23 PM
NeedHelp18
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
• Oct 26th 2008, 12:30 PM
Jhevon
Quote:

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)
• Oct 26th 2008, 03:45 PM
Krizalid
I'll assume that you've ever seen this symbol: $\sum.$

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Start with the base case $n=1$ and prove that it's true. Now, let's assume our proposition for $n=k,$ hence, it follows that $\sum\limits_{i=1}^{k}{(2i-1)}=k^{2}.$ We need to prove that $\sum\limits_{i=1}^{k+1}{(2i-1)}=(k+1)^{2}.$ Here's the proof:

\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 \\