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 numbersn.

1 + 3 + 5 + ...+ (2n- 1 ) =n^2

Printable View

- Oct 26th 2008, 12:23 PMNeedHelp18Mathematical 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 + ...+ (2*n*- 1 ) =*n*^2 - Oct 26th 2008, 12:30 PMJhevon
- Oct 26th 2008, 03:45 PMKrizalid
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}$