Show that the sum of the first positive odd integers is .

Show that the sum of the first positive even integers is

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- May 1st 2006, 08:29 PMc_323_hEasy problems
Show that the sum of the first positive odd integers is .

Show that the sum of the first positive even integers is - May 1st 2006, 08:34 PMThePerfectHacker
You have,

Or,

Regroup,

Using sum formulas,

Thus,

- May 1st 2006, 08:51 PMc_323_hQuote:

Originally Posted by**ThePerfectHacker**

why ? This evaluates to -1, and we're looking for the positive integers. - May 9th 2006, 11:37 AMAfterShock
(2n-1) for definition of odd; start the summation from 1...infinity.

Or, alternatively, 2n+1, and starting it from 0...infinity. - May 9th 2006, 07:55 PMThePerfectHacker
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. - May 28th 2006, 08:48 AMSoroban
Hello, c_323_h!

Quote:

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: .

Quote:

Show that the sum of the first positive even integers is

. . with first term and common difference

The sum is: .