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# What is the sum of positive integers from 1 to 1000?

What is the sum of positive integers from 1 to 1000?

How would I figure this out step by step?

### 13 Answers

- PuzzlingLv 71 year agoFavorite Answer
As others have noted, if the first number is 1 and the last number is 1000, the average is 500.5. All the other numbers will pair the same way (2 and 999, 3 and 998, etc.) with the same average.

If you have 1000 numbers that average to 500.5, the sum must be 500,500.

In general, the sum of the first n positive integers is:

n(n + 1)/2

For n=1000:

1000 * 1001 / 2

= 1000 * 500.5

= 500,500

And if you are only dealing with powers of 10, another shortcut is to take half the number and repeat it.

10 --> 55

100 --> 5050

1000 --> 500500

10000 --> 50005000

100000 --> 5000050000

etc.

Answer:

500,500

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- JimLv 71 year ago
Add the same numbers twice, the second row in reverse order

s = 1 + 2 + 3 + 4 + .... + 997 + 998 + 999 + 1000

s = 1000 + 999 + 998 + 997 + .... + 4 + 3 + 2 + 1

====================================

2s = 1001 + 1001 + 1001 etc 1000 times

2s = 1001 * 1000

2s = 1001000

s = 500,500 tada!

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- JohnLv 41 year ago
If you add them up one at a time you might find out something interesting and get the correct answer at the same time.

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- busterwasmycatLv 71 year ago
the simple way is to define the midpoint and generate pairs that make the same total. 500+501 is the same quantity as 1000+1. you would have 500 such pairs. so, the total must be 500x1001.

The only real trick is that some ranges have an odd number of items and the midpoint value has no equivalent (cannot be paired) so you have to add it separately. like in add 1 through 999; 500 is the median value. you would have 499x1000+500 as a total. or, from the other way, 500x1000-500. (double the median value to make an extra pair, and then subtract it, or its half-pair value)

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- Φ² = Φ+1Lv 71 year ago
The sum from 1 to 1 is 1 , which is 0.5 + 0.5

The sum from 1 to 10 is 1 + 2 + 3 + ... + 10 = 55

The sum from 1 to 100 is 1 + 2 + 3 + ... + 100 = 5050

Based on the pattern, the sum from 1 to 1000 is probably 500500

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- RayLv 71 year ago
One way is to treat it as an Arithmetic Series for which a = 1, d = 1 and n= 1000.

Sum = (n/2)[2a + (n - 1)d]

Sum = 500(2 + 99 x 1)

Sum = 500 x 101

Sum = 50500

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- Robert JLv 71 year agoReport
They missed a zero in the final value... 500500

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- geezerLv 71 year ago
So .. start with 1000

Then it's 1 + 999 = 1000

and .. 2 + 998 = 1000

3 + 997 = 1000

4 + 996 = 1000

etc etc until you get to 499 + 501

leaving yiou with the 500

It will take 500 steps to get here

so multiply 1000 by 500

and add the 500 which will be 'on it's own, in the middle'.

500x1000 + 500

500500

@ the comment by amania_r .. I DID ADD THE INITIAL 1000

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true, but not elegant. There is a formula.

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- Bone AloneLv 71 year ago
1 + 1000 = 1001

2 + 999 = 1001

3 + 998 = 1001

See a pattern developing.

By the time you get to 500 + 501, you would have used up all the numbers from 1 to 1000.

So 500(1001) gives you the answer.

true, but not elegant. There is a formula.

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- Anonymous1 year ago
There is a trick to this.

They can be laid out as a triangle.

*

**

***

Which is half a rectangle

****

****

****

The above is for 1 to 3. The sum is 3*4/2

Similarly your sum is 1000*1001/2

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