SOLUTION: how do you add 1+ 2+ 3+ +4 all the way up to say 200 or 300? Is there an equation to use for that?

Question 1142839: how do you add 1+ 2+ 3+ +4 all the way up to say 200 or 300? Is there an equation to use for that?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

Pick a small number for an example that you can see easily:

1+2+3+4+5+6+7+8+9+10 = ??

Group the numbers in pairs, like this:

(1+10)+(2+9)+(3+8)+(4+7)+(5+6)

The sum is now the sum of 5 pairs, with the sum of each pair being 11. So the sum of all the numbers is 5*11 = 55.

It works when the last number is odd also. For example,

1+2+3+4+5+6+7 = (1+7)+(2+6)+(3+5)+4

which gives you 3 1/2 pairs, each with a sum of 8; the sum is 8 times 3 1/2 = 4 times 7 = 28.

In general, if the last number is n, the sum of the first and last numbers is 1+n, and the number of pairs is n/2. That gives you the following general formula for the sum of the integers from 1 to n:

So for your two particular ending numbers,

1+2+3+...+200 = (200/2)(200+1) = 100*201 = 20100
1+2+3+...+300 = (300/2)(300+1) = 150*301 = 45150

Or, for a random odd last number,

1+2+3+...+79 = (79/2)(79+1) = (79/2)(80) = 79*40 = 3160
Answer by ikleyn(52780)   (Show Source): You can put this solution on YOUR website!
.
It is about arithmetic progressions . . .

For introductory lessons on arithmetic progressions see
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
in this site.

Find there answers to all your questions, in clear and compact form.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Arithmetic progressions".

Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.

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