Show that the sum of the first $\displaystyle n$ positive odd integers is $\displaystyle n^2$.
Show that the sum of the first $\displaystyle n$ positive even integers is $\displaystyle n^2+n$
You have,
$\displaystyle 1+3+5+...+(2n-1)$
Or,
$\displaystyle (2\cdot 0-1)+(2\cdot 1-1)+...+(2\cdot n -1)$
Regroup,
$\displaystyle 2(0+1+2+...+n)-(1+1+...+1)$
Using sum formulas,
$\displaystyle 2\cdot \frac{n(n+1)}{2}-n$
Thus,
$\displaystyle n^2+n-n=n^2$
oops, i meant to post this is the calculus section, but i guess it can apply to number theory.Originally Posted by ThePerfectHacker
why $\displaystyle (2\cdot 0-1)$? This evaluates to -1, and we're looking for the positive integers.
You can easily prove these two identities with
Mathematical Induction
I was just too lazy to use induction on these two problems. And decided to use a more elegant way.
Hello, c_323_h!
We have: .$\displaystyle 1 + 3 + 5 + \hdots + (2n-1)$Show that the sum of the first $\displaystyle n$ positive odd integers is $\displaystyle n^2$.
. . an Arithmetic Progression with first term $\displaystyle a = 1$ and common difference $\displaystyle d = 2$.
The sum of the first $\displaystyle n$ term is an A.P. is: .$\displaystyle S_n\;=\;\frac{n}{2}[2a + d(n-1)]$
So we have: .$\displaystyle S\;=\;\frac{n}{2}[2\cdot1 + 2(n-1)]\;=\;n^2$
We have another A.P. .$\displaystyle 2 + 4 + 6 + \hdots + 2n$Show that the sum of the first $\displaystyle n$ positive even integers is $\displaystyle n^2+n$
. . with first term $\displaystyle a = 2$ and common difference $\displaystyle d = 2.$
The sum is: .$\displaystyle S\;=\;\frac{n}{2}[2\cdot2 + 2(n-1)] \;= \;n(n + 1)$