# Thread: If the sum of the consecutive integers from -22 to x, inclusive, is 72

1. Sorry, add 23 to what?

2. Originally Posted by Potterpie
The sum of consecutive integers from -22 to 22, which is equal to 0.

3. 23.. =)

I'm lost. Sorry. You don't have to try.

4. Originally Posted by Potterpie
If the sum of the consecutive integers from -22 to x, inclusive, is 72
then what is the value of x?

this question is from my SAT study guide. Its difficulty is hard.

Thanks so much!

72 = -22 + (-21) + .... + 0 + ..... + (x - 1) + x .... (A)

Reverse the order:

72 = x + (x - 1) + ..... + 0 + ..... + (-21) + (-22) .... (B)

144 = (x - 22) + (x - 22) + ..... + (x - 22) + (x - 22).

There are 23 + x identical terms on the right. Therefore:

144 = (x - 22)(23 + x).

Solve for x (you can do that, right?) and keep the positive solution. I get x = 25.

5. Yes, I can solve for x.

Thanks. How did you know to reverse the order and add?

6. Originally Posted by Potterpie
Yes, I can solve for x.

Thanks. How did you know to reverse the order and add?
It's a trick I learned from Carl Gauss. If you want, you can read about him here: Carl Friedrich Gauss - Wikipedia, the free encyclopedia. He's a bit famous in certain circles .....

7. Okay, well, thanks. Still kind of confused but I get it a little better now.

8. Originally Posted by mr fantastic
It's a trick I learned from Carl Gauss. If you want, you can read about him here: Carl Friedrich Gauss - Wikipedia, the free encyclopedia. He's a bit famous in certain circles .....
He is awesome. I have his Disquisitiones Arithmeticae, very beautiful book

9. Originally Posted by Potterpie
Okay, well, thanks. Still kind of confused but I get it a little better now.
Another way of doing it - which has probably been mentioned, but I can't be bothered reading through the thread again - is to realise that if x = 22, then the sum is zero. Then note that 23 + 24 + 25 = 72. So x = 25.

No doubt that's how you're expected to do it.

Page 2 of 2 First 12