You have,
Or,
Regroup,
Using sum formulas,
Thus,
oops, i meant to post this is the calculus section, but i guess it can apply to number theory.Originally Posted by ThePerfectHacker
why ? 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: .Show that the sum of the first positive odd integers is .
. . an Arithmetic Progression with first term and common difference .
The sum of the first term is an A.P. is: .
So we have: .
We have another A.P. .Show that the sum of the first positive even integers is
. . with first term and common difference
The sum is: .